## Abstract

Temporal filters, the ability of postsynaptic neurons to preferentially select certain presynaptic input patterns, have been shown to be associated with the notion of information filtering and coding of sensory inputs. Their properties can be dynamically characterized as the transient responses to periodic presynaptic inputs. Short-term plasticity (STP) has been proposed to be an important player in the generation of temporal filters, but the response of postsynaptic neurons to presynaptic inputs depends on a collection of time scales in addition to STP’s, which conspire to create temporal filters: the postsynaptic time scales generated by the cellular intrinsic currents and the presynaptic time scales captured by the ISI distribution patterns. The mechanisms by which these time scales and the processes giving rise to them interact to produce temporal filters in response to presynaptic input spike trains are not well understood. We carry out a systematic modeling and computational analysis to understand how the postsynaptic low-, high- and band-pass temporal filters are generated in response to periodic presynaptic spike trains in the presence STP, and how the dynamic properties of these filters depend on the interplay of a hierarchy of processes: arrival of the presynaptic spikes, the activation of STP and its effect on the synaptic connection efficacy, and the response of the postsynaptic cell. The time scales associated with each of these processes operate at the short-term, single-event level (they are activated at the arrival of each presynaptic spike) and collectively produce the long-term time scales that determine the shape and properties of the filters. We develop a series of mathematical tools to address these issues for a relatively simple model where depression and facilitation interact only at the level of the synaptic efficacy change as time progresses and we extend our results and tools to account for more complex models that involve interactions at the STP level and multiple STP time scales. We use these tools to understand the mechanisms of generation of temporal filters in the postsynaptic cells in terms of the properties and dynamics of the interacting building blocks.

## 1 Introduction

The synaptic communication between neurons involves a multiplicity of interacting time scales and is affected by a number of factors including short-term plasticity [1–3], primarily involved in information filtering, long-term plasticity [4,5], involved in learning and memory [6], homeostatic plasticity [7], involved in the maintenance of function in the presence of changing environments, neuromodulation [8, 9], and astrocyte regulation [10, 11], in addition to the temporal properties of the presynaptic spikes, the intrinsic currents of the postsynaptic neurons and background noise activity.

Short-term plasticity (STP) (or short-term synaptic dynamics, STD) refers to the increase (synaptic facilitation) or decrease (synaptic depression) of the efficacy of synaptic transmission (strength of the synaptic conductance) over time in response to repeated presynaptic spikes with a duration in the range of milliseconds to minutes [1–3, 12, 13]. STP is ubiquitous both in invertebrate and vertebrate synapses, and has been shown to be important for neuronal computation [14–18] and information filtering (temporal and frequency-dependent) [12,19–38], and related phenomena such as burst detection [27,38], temporal coding and information processing [27, 28, 39–41], gain control [15, 42, 43], information flow [16, 36, 44] given the presynaptic history-dependent nature of STP, the prolongation of neural responses to transient inputs [45–47], the modulation of network responses to external inputs [48, 49], hearing and sound localization [50, 51], direction selectivity [52], attractor dynamics [53] (see also [43]), navigation (e.g., place field sensing) [30,33], vision (e.g., microsacades) [54], working memory [47,55] and decision making [56].

The notion of information filtering as the result of synaptic depression or facilitation is associated with the concept of temporal filters at the synaptic and postsynaptic levels, which are better understood in response to periodic presynaptic inputs [38, 57, 58] for a wide enough range of input frequencies. (See the schematic diagrams in Figs. 1-A and 2-A where *x* and *z* describe the evolution over time of synaptic depression and facilitation, respectively, and their product describes their combined activity assumed to be multiplicative.) In spite of the pervasiveness of STP and the consequences for information filtering, the mechanisms of generation of postsynaptic temporal filters in response to presynaptic input spikes are not well understood. In particular, it is not understood how the time scales associated with the dynamics of depression and facilitation interact with the presynaptic spike train time scale (interspike interval, ISI) and the postsynaptic membrane potential time scales to generate the resulting temporal filters, and how the dynamic properties of these temporal filters that govern their evolution over time depend on the said time scales.

Phenomenological models of synaptic depression and facilitation [21, 40, 57–65] describe the two phenomena as the evolution of two variables that abruptly decrease or increase, respectively, by a certain amount in response to each presynaptic spike and relax towards their steady-state during the presynaptic ISIs in an independent manner (as in Figs. 1-A and 2-A). At the arrival of each presynaptic spike, the synaptic function (*S*) is updated by an amount equal to the appropriate product of *x* and *z* (instead of a constant value as in equivalent models without STP).

The temporal filters (e.g., Figs. 1-A2), capturing the envelope of the depression and facilitation variables (e.g., Figs. 1-A1) develop as the result of the repetitive arrival of presynaptic spikes with rates of increase that depends on the value of the variables *x* and *z* at the arrival of each spike, which are controlled by both the time scales at which the relaxation of these variables to their steady-state occurs (e.g., the time constants *τ*_{dep} and *τ*_{fac} in Fig. 2-A1) and the presynaptic ISI.

These temporal filters can be captured by sequences of peak values (*X*_{n} and *Z*_{n} where *n* is the spike index) whose evolution are governed by difference equations and have time scales (e.g., *σ*_{dep} and *σ _{fac}*) and steady states (e.g., and ). We refer to them as a low-pass (

*X*

_{n}) and high-pass (

*Z*

_{n}) temporal filters, respectively. The synaptic update is the product Δ

*S*

_{n}=

*X*and the corresponding filter can have a transient peak, which we refer to as a band-pass temporal filter and involve an additional time scale (

_{n}Z_{n}*σ*

_{dep+fac}). These time scales depend on the single-event time scales (

*τ*

_{dep}and

*τ*

_{fac}) and the presynaptic ISI. In addition, the phenomenon of summation in a postsynaptic cell in response to presynaptic inputs develops an additional high-pass temporal filter (Fig. 2-B), which is independent of the ones described above, with a time scale (

*σ*

_{sum}) and steady state that depend on the membrane time scale of the postsynaptic cell (

*τ*

_{m}), the presynaptic ISI and the synaptic decay time constant (

*τ*).

_{dec}In this paper we use biophysically plausible (conductance-based) mathematical modeling and dynamical systems tools to systematically understand how the postsynaptic low-, high- and band-pass temporal filters are generated in response to presynaptic spike trains in the presence of STP and how the dynamic properties of these filters depend on the interplay of the hierarchy of processes (ranging from the arrival of the presynaptic spike trains, to the activation of STP to the activation of the synaptic function to the response of the postsynaptic cell), and the hierarchy of temporal filters with their associated time scales, and in particular how all this depends on the time scales of the building blocks (*τ*_{dep}, *τ*_{fac}, *τ*_{dec}, *τ*_{m} and the presynaptic ISI). The conceptual and mathematical framework we introduce to develop these ideas and identify the contribution of each of the components can be used to understand the filtering and coding properties of more complex scenarios involving more realistic description of the participating processes, non-periodic presynaptic spike trains (e.g., bursting patterns) and more complex networks.

## 2 Methods

### 2.1 Models

#### 2.1.1 Postsynaptic cell: leaky integrate-and-fire model

The current-balance equation for the post-synaptic cell is given by
where *t* is time (ms), *V* represents the voltage (mV), *C* is the specific capacitance (*μ*F/cm^{2}), *g*_{L} is the leak conductance (mS/cm^{2}), Iapp is the tonic (DC) current (*μ*A/cm^{2})), represents white noise (delta correlated with zero mean), and *I*_{syn} is an excitatory synaptic current of the form

Here *G*_{ex} is the maximal synaptic conductance (mS/cm^{2}), *E*_{ex} = 0 is the reversal potential for AMPA excitation, and the synaptic variable *S* obeys a kinetic equation of the form
where *τ*_{dec} (ms) is the decay time of excitation. Each presynaptic spike instantaneously raises *S* to some value Δ*S*_{n} which varies depending on the properties of the short-term dynamics (depression and/or facilitation) and defined below. We refer the reader to [59, 66] for additional details on biophysical (conductance-based) models.

#### 2.1.2 Presynaptic spike-trains

We model the spiking activity of the presynaptic cell as a spike train with presynaptic spike times *t*_{1}, *t*_{2}, … , *t*_{N} . We consider two types of input spike-trains: uniformly and Poisson distributed. The former is characterized by the interspike interval (ISI) of length Δ_{spk} (or its reciprocal, the spiking frequency *f*_{spk}) and the latter are characterized by the mean spiking rate (or the associated exponential distribution of ISIs).

#### 2.1.3 DA phenomenological model for short-term dynamics: synaptic depression and facilitation

This simplified phenomenological model has been introduced in [59]. It is relatively simpler than the well-known phenomenological MT model described below [58]. In particular, the depression and facilitation processes are independent (the updates upon arrival of each presynaptic spike are uncoupled). We use it for its tractability and to introduce some conceptual ideas.

The magnitude Δ*S* of the synaptic release per presynaptic spike is assumed to be the product of the depressing and facilitating variables
where
and

Each time a presynaptic spike arrives (*t* = *t*_{spk}), the depressing variable *x* is decreased by an amount *a*_{d} *x* (the release probability is reduced) and the facilitating variable *z* is increased by an amount *a*_{f} (1 − *z*) (the release probability is augmented). During the presynaptic interspike intervals (ISIs) both *x* and *z* decay exponentially to their saturation values *x*_{∞} and *z*_{∞} respectively. The rate at which this occurs is controlled by the parameters *τ*_{dep} and *τ*_{fac}. Following others we use *x*_{∞} = 1 and *z*_{∞} = 0. The superscripts “±” in the variables *x* and *z* indicate that the update is carried out by taking the values of these variables prior (_{−}) or after (^{+} the arrival of the presynaptic spike.

Fig. 1-A1 illustrates the *x*-, *z*- and *M* = *xz*-traces (defining the Δ*S*-traces) (curves of *x*, *z* and *M* as a function of time) in response to a periodic presynaptic input train for representative parameter values. (Note that *M* is a continuous variable, while Δ*S* is used for the update after the arrival of spikes, and Δ*S* = *R*^{−}*u*^{+} and Δ*S*_{n} is the sequence off peaks.)

#### 2.1.4 DA model in response to presynaptic inputs

##### Peak dynamics and temporal filters

By solving the differential equations (5)–(6) during the presynaptic ISIs and appropriately updating the solutions at *t* = *t*_{n} (occurrence of each presynaptic spike), one arrives at the following recurrent formula for the peak sequences in terms of the model parameters
and
where represents the lengths of the presynaptic ISIs.

Fig. 1-A2 illustrates the peak envelopes (curves joining the peak sequences for *X*_{n}, *Z*_{n} and Δ*S*_{n} = *X _{n}Z_{n}*, circles) for the parameter values in Fig. 1-A1. These are sequences indexed by the input spike number, which we calculate analytically below. As expected,

*X*

_{n}is a decreasing sequence (temporal low-pass filter) and

*Z*

_{n}is an increasing sequence (temporal high-pass filter). Their product (computed so that the peak of the product is the product of the peaks) exhibits a transient peak (temporal band-pass filter).

##### Steady-state frequency-dependent filters

For periodic inputs, Δ_{spk,n} is independent of *n* (Δ_{spk}) and eqs. (7)–(8) are linear 1D difference equations. Therefore both the sequences *X* and *Z* obey linear discrete dynamics (e.g., see [67]), decaying to their steady state values
and
as shown in Figs. 1 -B and -C (red and green).

#### 2.1.5 MT phenomenological model for short-term dynamics: synaptic depression and facilitation

This model was introduced in in [58] as a simplification of earlier models [1, 40, 68]. It is more complex and more widely used than the DA model described above [38, 69], but still a phenomenological model.

As for the DA model, the magnitude Δ*S* of the synaptic release per presynaptic spike is assumed to be the product of the depressing and facilitating variables:
where, in its more general formulation,
and

Each time a presynaptic spike arrives (*t* = *t*_{spk}), the depressing variable *R* is decreased by *R*^{−}*u*^{+} and the facilitating variable *u* is increased by *U* (1 − *u*^{−}). As before, the superscripts “±” in the variables *R* and *u* indicate that the update is carried out by taking the values of these variables prior (*R*^{−}, *u*^{−}) or after (*R*^{+}, *u*^{+}) the arrival of the presynaptic spike. In contrast to the DA model, the update of the depression variable *R* is affected by the value of the facilitation variable *u*^{+}.

Simplified versions of this model include making [21,57,58,63,65,70] and [38] where the updates do not discriminate between the values of the participating variables before or after the arrival of spikes

#### 2.1.6 MT model in response to presynaptic inputs

##### Peak dynamics and temporal filters

By solving the differential equations (12)–(13) during the presynaptic ISIs and appropriately updating the solutions at *t* = *t*_{n} (occurrence of each presynaptic spike), one arrives at the following recurrent formula for the peak sequences in terms of the model parameters
and

##### Steady-state frequency-dependent filters

As before, for presynaptic inputs Δ_{spk,n} is independent of *n* and these equations represent a system two 1D difference equations, which are now nonlinear. The steady-state values are given by
and

#### 2.1.7 Synaptic dynamics in response to periodic presynaptic inputs and a constant update

##### Peak dynamics

By solving the differential equation (3) for a constant value of Δ*S*_{n} = Δ*S* during the presynaptic ISIs and updating the solution at each occurrence of the presynaptic spikes at *t* = *t*_{n}, *n* = 1, … , *N*_{spk}, one arrives to the following discrete linear differential equation for the peak sequences in terms of the model parameters

##### Steady-states and frequency filters

The steady state values of (18) are given by

### 2.2 Numerical simulation

The numerical solutions were computed using the modified Euler method (Runge-Kutta, order 2) [71] with a time step Δ*t* = 0.01 ms (or smaller values of Δ*t* when necessary) in MATLAB (The Mathworks, Natick, MA).

## 3 Results

### 3.1 Temporal summation filters: the single-event and the (long-term) filter time scales coincide for constant updates

Temporal summation filters (SFs, Fig. 2-B3) refer to the long-term patterns generated in the response of a dynamical system to periodic stimulation by the accumulation of activity in response to the single events (responses to each stimulation event, local in time). Temporal summation synaptic filters are HPFs and naturally develop in the response of linear systems such as eq. (18) with a constant update (independent of the spike index) where the activity *S* decays during the presynaptic ISI and it is updated in an additive manner at the arrival of each presynaptic spike. If the quotient Δ_{spk}/*τ*_{dec} is finite, *S* will not be able to reach a small enough vicinity of zero before the next presynaptic spike arrives and then the *S*-peak envelope will increase across cycles. While summation does not require the presynaptic inputs to be periodic, the notion of filter we use here does.

The solution to equation (18) is given by
where *S*_{1} = Δ*S* (see Appendix A). This equation describes the temporal filter in response to the presy-naptic spike train. For technical purposes, one can extend eq. (20) to include the point (0, 0), obtaining

A further extension to the continuous domain yield which is the solution to

We use the notation *S*_{t} instead of *S*(*t*) to emphasize the origin of *S*_{t} as the continuous extension of a discrete sequence rather than the evolution of Eq. (3).

Together these results show that the time scale *τ*_{dec} for the single events (in response to each presynaptic spike is directly communicated to the temporal SF in a Δ_{spk}-independent manner. In other words, the dynamics of both the temporal SF and the single events are controlled by the same time constant *τ*_{dec}. While the time scale is independent of Δ_{spk}, the steady-state is Δ_{spk}-dependent.

### 3.2 Dynamics of the depression (*x*) and facilitation (*z*) variables and their interaction: emergence of temporal low-, high-and band-pass filters

#### 3.2.1 From single events (local in time) to temporal patterns and filters (global in time)

The dynamics of single events for the variables *x* (depression) and *z* (facilitation) are governed by eqs. (5)–(6) , respectively. After the update upon the arrival of a spike, *x* and *z* decay towards their saturation values *x*_{∞}(= 1) and *z*_{∞}(= 0), respectively.

The response of *x* and *z* to repetitive input spiking generates patterns for these variables in the temporal domain (e.g., Fig. 1-A1) and for the peak sequences *X*_{n} and *Z*_{n} in the (discrete) presynaptic spike-time domain (e.g., Fig. 1-A2). They consist of their transition from the initial peaks *X*_{1} and *Z*_{1} to and respectively, as *n* → ∞. The properties of these patterns depend not only on the parameters for the single events (*τ _{dep/fac}* and

*a*

_{d/f}, but also on the input frerquency

*f*

_{spk}(or the presynaptic ISI Δ

_{spk}) as reflected by eqs. (9)–(10) describing the peak-envelope steady-state values.

The peak envelope patterns have emergent, long-term time constants, for which we use the notation *σ*_{d} and *σ _{f}* (see also Section 3.3.2), which depend on

*τ*, Δ

_{dep/fac}_{spk}and

*a*

_{d/f}. The relationship between the long- and (single event) short-term time scales is more complex than straightforward identity between the two time scales discussed in the previous section where the single events were controlled only by the time constant. We discuss this in more detail later in the next section.

The Δ*S*_{n} envelope patterns combine these time scales in ways that involve different levels of complexity. Δ*S*_{n} patterns like the one shown in Fig. 1-A (blue) that exhibit a peak in the temporal domain (transient peak) have been referred to in the literature as temporal filters, borrowing the language from electric circuits in the frequency domain that exhibits resonance (including neuronal circuits). Here we refer to this type of patterns as temporal band-pass filters (BPFs) and to the Δ*S*_{n} patterns that are monotonically decreasing (Fig. 1-A, red) and increasing (Fig. 1-A, green) as temporal low- and high-pass filters (LPF and HPF), respectively.

#### 3.2.2 Depression- / facilitation-dominated regimes and transitions between them

In the absence of either facilitation or depression, Δ*S*_{n} temporal LPFs and HPFs reflect the presence of depressing or facilitating synapses, respectively. However, Δ*S*_{n} temporal LPFs and HPFs need not be generated by pure depression and facilitation but can reflect different balances between these processes where either depression (Fig. 1-B1) or facilitation (Fig. 1-B2) dominates.

It is instructive to look at the limiting cases. A small enough value of *τ*_{dep/fac} causes a fast recovery to the saturation value (*x*_{∞} or *z*_{∞}) and therefore the corresponding sequence (*X*_{n} or *Z*_{n}) is almost constant. In contrast, a large enough value of *τ*_{dep/fac} causes a slow recovery to the saturation value and therefore the corresponding sequence shows a significant decrease (*X*_{n}) or increase (*Z*_{n}) as the result of the corresponding underlying variables (*x* and *z*), being almost constant during the ISI. Therefore, when *τ*_{dep} ≫ *τ*_{fac}, depression dominates (Fig. 1-B1) and when *τ*_{dep} ≪ *τ*_{fac}, facilitation dominates (Fig. 1-B2).

In both regimes, the exact ranges depend on the input frequency *f*_{spk} (or the ISI Δ_{spk}). An increase in *f*_{spk} (decrease in Δ_{spk}) reduces the ability of *x* and *z* to recover to their saturation values within each presynaptic ISI, and therefore amplifies the depression and facilitation effects over the same time interval and over the same amount of input spikes (compare Figs. 1-C1 and -C2). Therefore, the different balances between *X*_{n} and *Z*_{n} as *f*_{spk} generate different types of Δ*S*_{n} patterns and may cause transitions between qualitatively different Δ*S*_{n} patterns.

### 3.3 Dependence of the (depression/facilitation) temporal low- and high-pass filter time scales on the single event time scales

#### 3.3.1 Communication of the single event time scales to the (long-term) history-dependent filters

Here we focus on understanding how the long-term time scale of the depressing and facilitating sequences *X*_{n} and *Z*_{n} (peak-envelope response to periodic presynaptic spikes) result from the interaction of the time constants for the single events (*τ*_{dep} and *τ*_{fac}) and the spiking input time scales Δ_{spk}.

From eqs. (7)–(8), which govern the evolution of the temporal sequences *X*_{n} and *Z*_{n}, their dynamics depend on the recovery time constants for the single events (*τ*_{dep} or *τ*_{fac}), the ISI of the presynaptic spikes Δ_{spk} and the update parameters (*a*_{d} and *a*_{f} , also for the single events). Standard methods (see Appendix A) applied to these linear difference equations for Δ_{spk,n} = Δ_{spk} (independent of *n*) yield
and
for *n* = 1, … , *N*_{spk}, where
and

The temporal patterns for the peak sequences *X*_{n} and *Z*_{n} (their transition from *X*_{1} and *Z*_{1} to and , respectively, as *n* → ∞) are spanned as the factors *Q*(*a*_{d}, *τ*_{dep})^{n−1} and *Q*(*a _{f} , τ_{fac}*)

^{n−1}approach zero (e.g., Figs. 3, gray). Because this occurs monotonically (in both cases), then the and patterns decrease and increase monotonically to and , respectively (e.g., Figs. 1 and 3, red and green dots). The convergence for

*a*

_{stp}< 1 is guaranteed by the fact that

*Q*(

*a*

_{stp},

*τ*

_{stp}) < 1. Biophysically plausible values of

*a*

_{d}and

*a*

_{f}are well within this range.

For each value of *n*, *Q ^{n}*

^{−1}is an increasing function of

*Q*and for each fixed value of

*Q*,

*Q*

^{n}^{−1}is a decreasing function of

*n*. Together, the larger

*Q*(

*a*

_{stp},

*τ*

_{stp}), the larger the sequence

*Q*

_{n}=

*Q*(

*a*

_{stp},

*τ*

_{stp})

^{n−1}and the slower

*Q*

_{n}=

*Q*(

*a*

_{stp},

*τ*

_{stp})

^{n−1}converges to zero. From eq. (26), all other parameters fixed,

*Q*(

*a*

_{stp},

*τ*

_{stp})

^{n−1}decreases slower the smaller Δ

_{spk}(larger

*f*

_{spk}) (compare Fig. 4 columns 1 to 3), the larger

*τ*

_{stp}(compare Fig. 4 rows 1 to 3) and the smaller

*a*

_{stp}(not shown in the figure).

The effective time scale of the sequence *Q*_{n} can be quantified by calculating the time it takes for *Q*_{n} to decrease from *Q*_{1} = 1 to 0.37 and multiply this number by Δ_{spk}. This yields
where *σ*_{Q} is expressed in decimal numbers and has units of time. By substituting *τ*_{stp} and *a*_{stp} by *τ*_{dep} and *a*_{d} (*X*_{n}) and by *τ*_{fac} and *a*_{f} (*Z*_{n}), respectively, we obtain the time scales *σ*_{dep} and *σ _{fac}* for the sequences

*X*

_{n}and

*Z*

_{n}. These time scales quantify the time it takes for

*X*

_{n}to decrease from

*X*

_{1}to and for

*Z*

_{n}to increase from

*Z*

_{1}to , respectively.

While the dynamics of *Q*(*a*_{stp}, *τ*_{stp}), *X*_{n} and *Z*_{n} depend on the quotient Δ_{spk}/τ_{stp} (the two interacting time scales for the single events), the long-term time scales (*σ*_{Q}, *σ*_{d} and *σ _{f}*) depend on these quantities in a more complex way (Fig.5-A). For the limiting case

*f*

_{spk}→ 0,

*σ*→

_{dep/fac}*τ*

_{dep/fac}. For the limiting case

*f*,

_{spk}*σ*→ 0. For values of

_{dep/fac}*f*in between (see Appendix B),

_{spk}*σ*and

_{dep}*σ*decrease between these extreme values (

_{fac}*dσ*Δ

_{dep/fac}/d_{spk}< 0). The dependence of

*σ*

_{dep}and

*σ*with

_{fac}*τ*and

_{dep}*τ*

_{fac}follows a different pattern since (

*dσ*> 0). Both

_{dep/fac}/dτ_{dep/fac}*σ*

_{dep}and

*σ*increase with

_{fac}*τ*

_{dep}and

*τ*

_{fac}, respectively. The dependence of

*σ*

_{dep}and

*σ*with

_{fac}*a*

_{d}and

*a*

_{f}follows a similar pattern (

*dσ*> 0). Details for these calculations are presented in the Appendix B.

_{d/f}/da_{d/f}One important feature is the dependence of the sequences *X*_{n} and *Z*_{n} on Δ_{spk} for fixed values of *τ*_{stp}. This highlights the fact the depression/facilitation-induced history-dependent filters are also frequency-dependent.

A second important feature is that multiple combinations of *τ*_{dep/fac} and Δ_{spk} give rise to the same sequence *X*_{n} and *Z*_{n}, which from eqs. (7)–(8) depend on the ratios

Constant values of *γ _{d}* and

*γ*generate identical sequences

_{f}*X*

_{n}and

*Z*

_{n}, respectively, which will be differently distributed in the time domain according to the rescaling provided by Δ

_{spk}, reflecting the long-term time scales

*σ*. This degeneracy also occurs for the steady state values and (9)–(10), generating the - and -profiles (curves of and as a function of the input frequency ). This type of degeneracies may interfere with the inference process of short-term dynamics from experimental data.

_{dep/fac}A third important feature is that the update values *a*_{d} and *a*_{f} that operate at the single events contribute to the long-term time scale for the filters and do not simply produce a multiplicative effect on the filters uniformly across events.

#### 3.3.2 Descriptive envelope model for short-term dynamics in response to periodic presynaptic inputs

Following a similar approach as the one developed in Section 3.1 (for the synaptic dynamics with constant updates, independent of the presynaptic number) we extend the peak sequences (24)–(25) to the continuous domain and

The dynamics for *X*_{t} and *Z*_{t} are 1D, but more complex than the linear 1D dynamics for *S*_{t} described in Section 3.1. In contrast to the dynamics of *S*_{t} in Section 3.1, the filter and single-event dynamics for depression and facilitation are different. In particular, the time constant for the depression and facilitation filters involves the parameters *a*_{d} and *a*_{f} , respectively, which are absent in the single-event dynamics.

Here we propose a further simplification consisting of a descriptive model that captures the response of the depression and facilitation variables to periodic presynaptic inputs. It is based on the observation of the shapes of *X*_{n} and *Z*_{n} in Fig. 1, which suggests an exponential decay of these sequences to and similar to the evolution of *S* in response to con periodic presynaptic inputs and a constant update given by (22), even though the form of *X*_{t} and *Z*_{t} indicate this evolution is not strictly exponential. From eqs. (24)–(26) for the DA model we know the evolution is not strictly exponential, but it involves additional factors in *Q*(*a*_{d}, *τ*_{dep})^{n−1} and *Q*(*a _{f} , τ_{fac}*)

^{n−1}. Therefore, the model we describe here is simpler than the DA model described above. Although is simply descriptive, it allows us to understand certain ideas of how temporal (history-dependent) filters interact.

Assuming exponential decay, the peak dynamics can be captured by the following envelope functions
and
for *X* and *Z*, respectively where we are using the same notation for the time constants as for the DA model. The parameters can be computed from these sequences by matching the initial values (*F* (*t*_{1}) = *X*_{1} and *G*(*t*_{1}) = *Z*_{1}), the steady steady state values ( and ) and intermediate values (*t _{c}, X_{c}*) and (

*t*) chosen to be in the range of fastest increase/decrease of the corresponding sequences (∼ 50% of the gap between the initial and steady state values). This gives

_{c}, Z_{c}Fig. 1-B (solid) shows the plots of *F* (red), *G* (green) and *H* = *FG* (blue) superimposed with the *X*_{n}-, *Z*_{n}- and Δ*S*_{n}-sequence values. The error between the sequences and the envelope curves (using a normalized sum of square differences) is extremely small in both cases, consistent with previous findings [67].

#### 3.3.3 Communication of the single event time scales to the (long-term) history-dependent filters: envelope descriptive model

Eqs. (35) provide the time scales *σ*_{d} and *σ _{f}* for the functions

*F*(

*t*

_{k}) and

*G*(

*t*

_{k}) (32)–(33) in terms of

*τ*

_{dep/fac}, Δ

_{spk},

*a*

_{d/f}and the other model parameters for the sequences

*X*

_{n}and

*Z*

_{n}through their steady-states and values (9)–(10). In order to estimate these time scales for

*X*

_{n}and

*Z*

_{n}we compute the values of

*t*

_{c}and

*X*

_{c}and

*Z*

_{c}.

For *f*_{spk} → 1, both *X*_{n} and *Z*_{n} are almost constant, since the *x*(*t*) and *z*(*t*) have enough time to recover to their steady state values before the next input spike arrives, and therefore *σ*_{d} ≫ 1 and *σ _{f}* ≫ 1. In contrast, for

*f*

_{spk}≫ 1,

*x*(

*t*) and

*z*(

*t*) have little time to recover before the next input spike arrives and therefore they rapidly decay to their steady state values. In the limit of

*f*

_{spk},

*σ*

_{d}=

*σ*= 0. In between these two limiting cases,

_{f}*σ*

_{d}and

*σ*are decreasing functions of

_{f}*f*

_{spk}(Figs. 4-A1 and -A2). For fixed-values of

*f*

_{spk}, both

*σ*

_{d}and

*σ*are increasing functions of

_{f}*τ*

_{dep}and

*τ*

_{fac}, respectively.

### 3.4 Dynamics of the interplay of the depression and facilitation variables x and *z*: Emergence of temporal band-pass filters for Δ*S*_{n} = *X*_{n}*Z*_{n}

Under certain conditions, the interplay of depression and facilitation generates temporal band-pass filters in response to periodic inputs for Δ*S* = *x z* (Fig. 1-A), which are captured by the sequence Δ*S*_{n} = *X _{n}Z_{n}* (Fig. 1-A2 and -C2). A band pass filter requires and the existence of a spike index

*r*such that . This in turn requires that the two time constants

*τ*

_{dep}and

*τ*

_{fac}are such that they create the appropriate balance between the two temporal filter time constants

*σ*

_{dep}and

*σ*(or

_{fac}*σ*

_{d}and

*σ*) to support a band-pass filter for the product of the low- and high-pass temporal filters.

_{f}#### 3.4.1 Descriptive rules for the generation of temporal (envelope) band-pass filters from the interplay of the temporal (envelope) low- and high-pass filters

From a geometric perspective, temporal band-pass filters in response to periodic presynaptic inputs arise as the result of the product of two exponentially increasing and decreasing functions both decaying to-wards their steady-state (e.g., Fig. 6). At the descriptive level, this is captured by the temporal envelope functions (*F* , *G* and *H* = *FG*) discussed above whose parameters are not the result of a sequence of single events but are related to the biophysical model parameters by comparison with the developed temporal filters. These functions provide a geometric/dynamic way to interpret the generation of temporal filters in terms of the properties of depression (decreasing functions) and facilitation (increasing functions) in response to periodic inputs, although this interpretation uses the developed temporal filters and therefore is devoid from any biophysical mechanistic interpretation.

In order to investigate how the multiplicative interaction between *F* (*t*) and *G*(*t*) given by eqs. (32)–(33) give rise to the temporal band-pass filters *H* = *FG*, we consider a rescaled version of these functions
and
where *B* = 1 and

The function *G* transitions from *G*(0) = 1 − *C* to lim_{t→∞} *G*(*t*) = 1 with a fixed time constant (Fig. 7, green curves). The function *F* transitions from *F* (0) = 1 to lim_{t→∞} *F* (*t*) = *A* with a time constant *η* (Fig. 7, red curves). It follows that *H* transitions from *H*(0) = 1 − *C* to lim_{t→∞} *H*(*t*) = *A B* = *A* (Fig. 7, blue curves). A temporal band-pass filter is generated if *H* raises above *A* for a range of values of *t*. This requires *F* to decay slow enough so within that range *H* = *FG > A* (Fig. 7-A) or *A* to be small enough (Fig. 7-B). In fact, as *A* decreases, the values of *η* required to produce a band-pass temporal filter increases (compare Fig. 7-A2 and -B2).

Changes in the parameter *B* in (37) affect the height of the band-pass temporal filter, but not the generation mechanism described above. However, for certain ranges of parameter values *H* is a temporal low-pass filter (not shown).

#### 3.4.2 Mechanisms of generation of temporal band-pass filters

From a biophysically grounded mechanistic perspective, illustrated by Figs. 6 (see also Fig. 3-A), band-pass filters (panels C, D, E) provide a transition mechanism from low-pass filters for low enough input frequencies (panels A and B) to high-pass filters for high-enough frequencies (panel F, which is strictly not a high-pass filters, but it is effectively so for the time scale considered). The depression and facilitation time constants *τ*_{dep/fac} modulate these transitions through the dependence of *X*_{n} and *Z*_{n} on the quotients Δ_{spk}/τ_{dep/fac} (Fig. 8-B to -D).

By design, the geometric/dynamic mechanism described in the previous section is based on the assumption that the parameters *A*, *B*, *C*, *σ*_{d} and *σ _{f}* (or

*η*=

*σ*) are free (and independent). There is no notion of single event dynamics within this framework. However, from (9)–(10) and (34)–(35) these parameters are not independent in the realistic model (e.g., they all depend on the presynaptic input frequency

_{d}/σ_{f}*f*

_{spk}or period Δ

_{spk}). Simple calculations show that is a decreasing function of

*f*

_{spk}(e.g., Fig. 8-A1) (increasing function of Δ

_{spk}) and is an increasing function of

*f*

_{spk}(e.g., Fig. 8-A2) (decreasing function of Δ

_{spk}). The dependence of

*η*with

*f*

_{spk}is more complex and can be either increasing (e.g., Fig. 8-A3) or decreasing (not shown).

The behavior of and *η* in Fig. 8 illustrates that Δ*S*_{n} temporal band-pass filters can be generated for these model parameters as *f*_{spk} increases (Figs. 8-B to -D), and it will be more pronounced as *τ*_{dep} = *τ*_{fac} increases (compare panels B1 to D1 and B2 to D2). Note that a Δ*S*_{n} temporal band-pass filter exists for *τ*_{dep} = *τ*_{fac} = 300, but not for *τ*_{dep} = *τ*_{fac} = 100, although *τ _{dep}/τ_{fac}* = 1. Because of the dependence of

*X*

_{n}and

*Z*

_{n}on the quotients Δ

_{spk}/τ

_{dep/fac}, Δ

*S*

_{n}temporal band-pass filters can be generated by increasing values of

*τ*

_{dep},

*τ*

_{fac}or both (not shown).

### 3.5 Δ*S*_{n}-band pass filters require a third (emergent) time scale whose contribution is independent from the low- and high-pass filters’ time scales

The linear one-dimensional dynamics for both depression and facilitation at the (short-term) single event level generate linear one-dimensional discrete dynamics at the (long-term) temporal (low- and high-pass) filters level where the long-term time constants (*σ*_{d} and *σ _{f}*) depend on the short-term time constants (

*τ*

_{dep}and

*τ*

_{fac}) and the input frequency (Δ

_{spk}). From the discrete dynamics point of view, the temporal band-pass filters (BPFs) obtained as the product of the temporal LPF and HPF are overshoots where the sequence evolves by peaking at a higher value than the steady-state (without exhibiting damped oscillations). Overshoots require at least two-dimensional dynamics (generated by a difference equation where each value of the resulting sequence depends on the previous two) with the appropriate time scales. In [67] we showed that in certain circumstances the generation of temporal BPFs requires three-dimensional dynamics. Here we investigate how the time scales giving rise to temporal BPF depend on the time scales of the temporal LPF and HPF and these of the corresponding single events.

#### 3.5.1 Mechanistic DA model

The dynamics of the three last terms in eq. (53) are governed by *Q*(*a*_{d}, *τ*_{dep})^{n−1}, *Q*(*a _{f}, τ_{fac}*)

^{n−1}and [

*Q*(

*a*

_{d},

*τ*

_{dep}),

*Q*(

*a*)]

_{f}, τ_{fac}^{n−1}, respectively, and as

*n*→ ∞. The first two are given eq. (26) with

*τ*

_{stp}substituted by

*τ*

_{dep}and

*τ*

_{fac}, and

*a*

_{stp}substituted by

*a*

_{d}and

*a*

_{f}, accordingly. The corresponding time scales are given by eq. (28) with the same substitutions. The last one is given by

The product [*Q*(*a*_{d}, *τ*_{dep}) *Q*(*a _{f} , τ_{fac}*)]

^{n−1}cannot be expressed in terms of a linear combination of

*Q*(

*a*

_{d},

*τ*

_{dep})

^{n−1}and

*Q*(

*a*)

_{f}, τ_{fac}^{n−1}and therefore the four terms in (53) are linearly independent.

The time scale associated to the fourth terms in (53) can be computed from (54) (as we did before) by calculating the time it takes for [*Q*(*a*_{d}, *τ*_{dep}) *Q*(*a _{f} , τ_{fac}*)]

^{n−1}to decrease from [

*Q*(

*a*

_{d},

*τ*

_{dep})

*Q*(

*a*)]

_{f}, τ_{fac}^{n−1}from 1 (is value for

*n*= 1) to 0.37 and multiply this number by Δ

_{spk}. This yields

This long-term time scale has similar properties as *σ*_{dep} and *σ*_{fac}. In particular, for *f*_{spk} → 0, *σ*_{dep+fac} → *σ*_{dep}*σ*_{fac}/(*σ*_{dep} + *σ*_{fac}). For *f _{spk}* → ∞,

*σ*

_{dep}+

*f*

_{ac}→ 0. For values of

*f*

_{spk}in between,

*σ*

_{dep}

_{+fac}decrease between these extreme values. This is illustrated in Fig. 5-B along the time other two times scales,

*σ*

_{dep}and

*σ*(Fig. 5-B).

_{fac}To say that the three time scales (*σ*_{dep}, *σ _{fac}* and

*σ*

_{dep+fac}) are independent is equivalent to state that the dynamics is three dimensional, while the dynamics of the depression and facilitation sequences are one-dimensional. It also means that erasing one of the terms in Δ

*S*

_{n}is equivalent to projecting the three-dimensional signal into a two-dimensional space with the consequent loss of information if it does not provide a good approximation to the original signal, and this loss of information may in principle be the loss of the band-pass filter (overshoot). On the other hand, two-dimensional systems are able to produce overshoots. So the question arises of whether the signal Δ

*S*

_{n}without the last term (that combines the time scales of the two filters

*X*

_{n}and

*Z*

_{n}) preserves the temporal band-pass filter and, if yes, under what conditions.

In order to test the necessity of this third time scale for the generation of temporal band-pass filters, we consider the “cut” sequence
where and . For *a*_{d} = *a*_{f} and *τ*_{dep} = *τ*_{fac}, *Q*(*a*_{d}, *τ*_{dep}) = *Q*(*a _{f} , τ_{fac}*), and therefore Δ

*S*has the same structure as

_{cut,n}*X*

_{n}and

*Z*

_{n}, and therefore Δ

*S*cannot generate a temporal band-pass filter regardless of the value of Δ

_{cut,n}_{spk}. This includes the examples presented in Fig. 3. For other parameter values, standard calculations show that the parameter ranges for which Δ

*S*shows a peak are very restricted and when it happens, they rarely provide a good approximation to the temporal band-pass filter exhibited by Δ

_{cut,n}*S*

_{n}. Fig. 5-C illustrates this for a number of representative examples.

#### 3.5.2 Descriptive envelope model

Here we address similar issues in the descriptive models described in the previous section. We consider the function
which approximates the behavior of Δ*S*_{n}. From (32)–(33),
where
and *σ*_{d} and *σ _{f}* are given by (35). Note that

*σ*

_{d},

*σ*and

_{f}*σ*

_{d}

_{+f}are different quantities from

*σ*

_{dep},

*σ*and

_{fac}*σ*

_{dep}

_{+fac}discussed above.

This shows, from a different perspective than in the previous section, that the dynamics of Δ*S*_{n} involves three time scales: the two times scales discussed above (*σ*_{d} and *σ _{f}*) and the combined time scales

*σ*

_{d}

_{+f}(Fig. 4-A3).

Together, these results and the results from the previous section shows that while *X*_{n} and *Z*_{n} are generated by a 1D (linear) discrete dynamical systems (1D difference equations), Δ*S*_{n} is generated by a 3D (linear) discrete dynamical system (3D discrete difference equation). Under certain conditions, a 2D (linear) discrete dynamical system will produce a good approximation. Δ*S*_{n} is able to exhibit a band-pass filter because of the higher dimensionality of its generative model.

In order to understand the contribution of the combined time scale *σ*_{d}_{+f}, we look at the effect of cutting the fourth term in H (43). We call this function *H*_{cut}. Fig. 4-B shows that *H*_{cut} does not approximate Δ*S*_{n} well during the transients (response to the first input spikes) and fails to capture the transient peaks and the temporal band-pass filter properties of Δ*S*_{n}. This discrepancy between Δ*S*_{n} (or H) and *H*_{cut} is more pronounced for low than for high input frequencies. In fact , while *H*(*t*_{1}) = *a*_{f}. The question remains of whether there could be a 2D (linear) dynamical system able to reproduce the temporal filters for Δ*S*_{n} with (emergent) time scales different from *σ*_{d} and *σ _{f}* . In other words, whether Δ

*S*

_{n}can be captured by a function of the form where

*h*

_{0},

*h*

_{1},

*h*

_{2},

*σ*

_{1}and

*σ*

_{2}are constants where by necessity,

### 3.6 Interplay of short-term synaptic and cellular postsynaptic dynamics: temporal BPFs generated within and across levels of organization

Standard models of synaptic transmission consider synaptic currents such as eq. (2). The synaptic function *S*, which controls the synaptic efficacy, obeys a first linear kinetic equation such as eq. (92) (*κ* = Δ*S* = 1) in the Appendix C [72, 73] and saturates at *S* = 1. The raise time *τ _{rse}* is typically very fast, at least an order of magnitude faster than the decay time (

*τ*

_{dec}) and therefore some simplified models consider an instantaneous increase right after the arrival of a presynaptic spike and a subsequent decay governed by an eq. such as eq. (3) with Δ

*S*

_{n}= 1 for all

*n*. Consistent with the saturation principle, the update should be an increment to

*S*= 1. From this perspective, the dynamics discussed in Sections 2.1.7 and 3.1 would not be strictly applicable for synaptic functions since the update considered there is an increment by Δ

*S*= 1 (and not to Δ

*S*= 1). More specifically, temporal summation is not expected for the synaptic variable

*S*. On the other hand, the model with an update by Δ

*S*= 1 (Sections 2.1.7 and 3.1) is applicable as a simplified model of post-synaptic potential (PSP) response to presynaptic inputs if

*τ*

_{dec}is interpreted as the membrane potential decay time.

Earlier models of synaptic dynamics consider the PSP peak sequence to be proportional to Δ*S*_{n} = *X _{n}Z_{n}* [21, 58, 63]. This approach does not take into consideration the dynamic interaction between the synaptic function

*S*and the membrane potential. While the synaptic dynamics with the update to

*S*= 1 reproduces the peak pattern of Δ

*S*

_{n}, in these models, the PSP response does not reflect the effects of the membrane time scale. Later models consider the synaptic function

*S*embedded in the current-balance equation and obeying equations of the form (92) where the rate of

*S*increase (rise time) is proportional to Δ

*S*

_{n}[74] or a similar model with

*κ*= Δ

*S*= 1, but with the saturation value equal to a function of Δ

*S*

_{n}[64] (see Appendix C). The presence of additional times scales further in the line (e.g., membrane potential time scale) gives rise to the phenomenon of temporal summation and the associated HPF in response to periodic synaptic inputs in the absence of STD (see Fig. 2-B).

In this paper, we follow a dual approach to investigate the mechanisms of generation of PSP temporal filters in response to periodic inputs. We first consider postsynaptic dynamics governed by eq. (2) with *S* increased by Δ*S*_{n} and extensions where the decay time reflects the membrane potential dynamics. This model has the advantage of being analytically solvable and it allows us to understand the effects of the filtering (depression, facilitation and summation) time scales in terms of the single event time constants (*τ*_{dep}, *τ*_{fac} and *τ*_{dec}). Then, we use the more biophysically realistic approach by using eqs. (1)–(3) with *S* increased to Δ*S*_{n} at the arrival of each presynaptic spike. In this model the interplay between the synaptic input and the postsynaptic cell is multiplicative (nonlinear).

#### 3.6.1 PSP temporal filters in the simplified model

We use eq. (3) with an update by Δ*S*_{n}. Here *S* is interpreted as an intermediate between the synaptic and the postsynaptic response with longer time scales than the standard AMPA and GABA_{A} synapses, reflecting the membrane potential time scales of postsynaptic cells. Our goal is to understand how the (global) time scales of the *S*-response patterns to periodic inputs depend on the time constant *τ*_{dec} and the depression and facilitation time scales *τ*_{dep} and *τ*_{fac} through the Δ*S*_{n} filter time scales *σ*_{dep} and *σ _{fac}*.

In the absence of depression and facilitation (*τ _{dep}, τ_{fac}* 0), Δ

*S*

_{n}is constant across cycles and

*S*generates temporal summation HPFs as described in Section 3.1 (Fig. 2-B) with

*σ*

_{sum}=

*τ*

_{dec}. We use the notation for the corresponding sequences. In the presence of either depression or facilitation, the update Δ

*S*

_{n}is no longer constant across cycles and therefore, the STD LPFs and HPFs interact with the summation HPFs. From a different perspective, the inputs to

*S*are no longer constant across cycles but modulated by STD.

By solving the differential equation (3) where *S* is increased by Δ*S*_{n} at the arrival of each spike (*t* = *t*_{n}, *n* = 1, … , *N*_{spk}) one arrives to the following discrete linear differential equation for the peak sequences in terms of the model parameters

The solution to eq. (47) is given by the following equation involving the convolution between the STP input Δ*S*_{n} and an exponentially decreasing function
with *S*_{1} = Δ*S*_{1} = *a*_{f} . The evolution of *S*_{n} is affected by the history of the STP inputs Δ*S*_{n} weighted by an exponentially decreasing function of the spike index and a coefficient

The steady state is given by
where given by (9) and (10). Note that Eq. (19) is a particular case of eq. (50) when Δ*S*_{n} is a constant sequence (no STD).

Both *S*_{n} and depend on Δ_{spk} and the time constants *τ*_{dep}, *τ*_{fac} and *τ*_{dec} through the quotients (29) and (49). Therefore, here we consider temporal patterns for a fixed-value of Δ_{spk}. The temporal patterns for other values of Δ_{spk} will be temporal compressions, expansions and height modulations of these baseline patterns. The values of *τ*_{dep}, *τ*_{fac} and *τ*_{dec} used in our simulations should be interpreted in this context.

For the limiting case *τ*_{dec} → 0, *S*_{n} → Δ*S*_{n} for 1 = 2, … , *N*_{spk}. The *S*-temporal filter reproduces (is equal to or a multiple of) the Δ*S* as in [21, 58, 63]. For the limiting case *τ*_{dec} → ∞, reflecting the lack of convergence of the sum in eq. (48). As the presynaptic spike number increases, the *S*-temporal filter reproduces the *S*^{0}-temporal filter since the Δ*S*_{n} → Δ^{−}*S*. In the remaining regimes, changes in *τ* affect both the steady state and the temporal properties of *S* in an input frequency-dependent manner as the *S*-temporal filter transitions between the two limiting cases. The former will be the object of a separate study. Here we focus on the temporal filtering properties. For visualization purposes and to compare the (global) time constants of the *S*, Δ*S* and *S*^{0} temporal filters, we present the *S*_{n}, Δ*S*_{n} and curves normalized by their maxima.

##### Emergence of *S*_{n} temporal BPFs: Interplay of synaptic depression (LPF) and postsynaptic summation (HPF)

In the Δ*S*_{n} facilitation-dominated regime (Fig. 9-A), the PSP *S*_{n} patterns result from the interaction between two HPFs, the Δ*S*_{n} facilitation and the summation ones. The filter time constant increases with increasing values of *τ*_{dec} reflecting the dependence of the summation (global) time constant *σ*_{sum} with *τ*_{dec}.

In the Δ*S*_{n} depression-dominated regime (Fig. 9-B), temporal PSP *S*_{n} BPFs emerge as the result of the interaction between the Δ*S*_{n} depression LPF and a HPF for intermediate values of *τ*_{dec} (Fig. 9-B2). *S*_{n} BPFs are not present for small enough values of *τ*_{dec} (Fig. 9-B1) since this would require Δ*S*_{n} to be a BPF, and are also not present for large enough values of *τ*_{dec} (Fig. 9-B3) since this would require to be a BPF. As for the depression/facilitation BPFs discussed above, the *S*_{n} BPFs are a balance between the two other filters and emerge as *S*_{n} transitions in between them as *τ*_{dec} changes.

##### Dislocation of the (output) *S*_{n} temporal BPF from the (input) Δ*S*_{n} temporal BPFs

In this scenario, a depression-facilitation Δ*S*_{n} BPF is generated at the synaptic level and interacts with the summation HPF (Fig. 9-C). The Δ*S*_{n} BFP evokes a PSP *S*_{n} BFP for low enough values of *τ*_{dec} (Fig. 9-C1). As *τ*_{dec} increases, the *S*_{n} pattern transitions to the HPF (Fig. 9-C3). As this transition happens, the *S*_{n} BPF moves to the right and increases in size before entering the summation-dominated HPF regime. While the PSP *S*_{n} BPF is inherited from the synaptic regime, its structure results from the interplay of the synaptic BPF and PSP temporal summation.

#### 3.6.2 Biophysically realistic models reproduce the above PSP temporal filters with similar mechanisms

Here we test whether the results and mechanisms discussed above remain valid for the voltage response to periodic presynaptic inputs by using the more realistic, conductance-based model (1)–(3). Here *S* has its original interpretation as a synaptic function with relatively small time constants, consistent with AMPA and GABA_{A} synapses. The value of *S* is updated to Δ*S*_{n} = *X _{n}Z_{n}* upon the arrival of each spike (at a time

*t*

_{n}), capturing the effects of synaptic depression and facilitation produced by the DA model. Because the interaction between the synaptic variable

*S*and

*V*are multiplicative, the model is not analytically solvable.

The *V* temporal patterns generated by this model (Fig. 10) are largely similar to the ones discussed above (Fig. 9) and are generated by similar mechanisms described in terms of the interplay of the membrane potential time scale *τ*_{m} (= *C/gL*) and the depression/facilitation time scales (*τ*_{dep} and *τ*_{fac}) through the (global) Δ*S*_{n} filter time scales (*σ*_{dep} and *σ _{fac}*). Because

*τ*

_{dec}is relatively small,

*S*largely reproduces the temporal properties of the Δ

*S*

_{n}pattern. As before, refer to the voltage response to presynaptic periodic inputs in the absence of STD (

*S*is updated to Δ

*S*

_{n}constant). Fig. 10-A illustrates the generation of

*V*temporal BPFs as the result of the interaction between synaptic depression (LPF) and postsynaptic summation (HPF). Fig. 10-B illustrates the dislocation of postsynaptic BPF inherited from the synaptic input Fig. 10-A. We limited our study to realistic values of

*τ*

_{m}.

### 3.7 Interplay of multiple depression and facilitation process with different time scales

In the models discussed so far both STD and STF involve one time scale (*τ*_{dep} and *τ*_{fac}) that governs the evolution of the corresponding variables (*x* and *z*) in between presynaptic spikes. These single-event time scales are the primary component of the temporal filter time scales (*σ*_{d} and *σ _{f}*) that develop in response to the periodic repetitive arrival of presynaptic spikes.

Here we extend these ideas to include scenarios where depression and facilitation involve more than one time scale. We interpret this as the coexistence of more than one depression or facilitation process whose dynamics are governed by a single single-event time scale each. Similar to the standard model, the independent filters that develop in response to the presynaptic spike train inputs interact to provide an input to the synaptic dynamics. In principle, this interaction may take various forms. Here, for exploratory purposes and to develop ideas, we consider a scenario where the processes of the same type (depression or facilitation) are linearly combined and the interaction between depression and facilitation is multiplicative as for the single depression-facilitation processes. We refer to it as the distributive or cross model. In the Appendix D we discuss other possible formulations. The ideas we develop here can be easily extended to more STD processes.

#### 3.7.1 The cross (distributive) model

More specifically, the depression and facilitation variables *x*_{k} and *z*_{k}, *k* = 1, 2 obey equations of the form (5)–(6) with parameters *τ _{dep,k}*,

*τ*,

_{fac,k}*a*and

_{d,k}*a*for

_{f,k}*k*= 1, 2. The evolution of these variables generate the sequences

*X*and

_{k,n}*Z*(

_{k,n}*k*= 1, 2) given by 24)–(25) with the steady-state values and given by (9)–(10). For simplicity, we consider

*a*

_{d,}_{1}=

*a*

_{d,}_{2}and

*a*

_{f,}_{1}=

*a*

_{f,}_{2}(and omit the index

*k*) so the differences between two depression or facilitation filters depend only on the differences of the single-event time constants. This can be easily extended to different values of these parameters for the different processes. In what follows, we will not specify the range of the index

*k*= 1, 2 unless necessary for clarity.

In the cross (or distributive) model, the variable *M* is given by
where *η _{d,}*

_{1}+

*η*

_{d,}_{2}= 1 and

*η*

_{f,}_{1}+

*η*

_{f,}_{2}= 1. Correspondingly, the synaptic update is given by for

*n*= 1, … ,

*N*

_{spk}. This model allows for all possible interactions between the participating depression and facilitation processes. It reduces to the single depression-facilitation process for

*η*

_{d,}_{2}=

*η*

_{f,}_{2}= 0 (or

*η*

_{d,}_{1}=

*η*

_{f,}_{1}= 0) and allows for independent reductions of depression and facilitation by making

*η*

_{d,}_{2}= 0 or

*η*

_{f,}_{2}= 0, but not both simultaneously.

From (24)–(26),
and
for *k* = 1, 2 with
where for use the notation and to refer to the first elements in the sequences, which, for simplicity, are assumed to be independent of *k*.

We use the notation where after algebraic manipulation, and

#### 3.7.2 Depression (Δ*S*_{dep,n}), facilitation (Δ*S*_{fac,n}) and temporal filters

_{dep,n}

_{fac,n}

The history-dependent temporal filter Δ*S _{dep,n}* transitions from to as

*n*→ ∞, and Δ

*S*transitions from to as

_{fac,n}*n*→ ∞. Because the individual filters are monotonic functions, the linearly combined filters represented by the sequences Δ

*S*and Δ

_{dep,n}*S*are also monotonic functions lying in between the corresponding filters for the individual filter components (Figs. 11-A1 and 12-A1for depression and Figs. 11-A2 and 12-A2 for facilitation). As a consequence, the filters also lie in between the product of the corresponding individual filter components Δ

_{fac,n}*S*

_{1,n}and Δ

*S*

_{2,n}(Figs. 11-A3 and Figs. 12-A3). In these figures, all parameter values are the same except for

*τ*

_{dep,}_{1}and

*τ*

_{fac,}_{1}, which are

*τ*

_{dep,}_{1}=

*τ*

_{fac,}_{1}= 100 in Fig. 11 and

*τ*

_{dep,}_{1}=

*τ*

_{fac,}_{1}= 10 in Fig. 12. In both figures, the values of the facilitation time constants are

*τ*

_{dep,}_{2}=

*τ*

_{fac,}_{2}= 1000.

In Fig. 11-A3 both Δ*S*_{1,n} and Δ*S*_{2,n} are temporal BPFs peaking almost at the same time. Consequently is also temporal BPFs lying strictly in between the individual ones and peaking almost at the same time. In Fig. 12-A3, in contrast, Δ*S*_{1,n} is a temporal HPF, while Δ*S*_{2,n} is a temporal BPF. The resulting is also a temporal BFP, but the two temporal BPFs peak at different times.

#### 3.7.3 Communication of the single event time scales to the history-dependent filters

In Section 3.3.2 we developed a descriptive envelope model for STD in response to periodic presynaptic inputs consisting of the functions *F* (*t*) for depression, *G*(*t*) for facilitation, and *H*(*t*) = *F* (*t*)*G*(*t*) for the synaptic update, given by eqs. (32)–(34) and (43). By approximating the model parameters using the results of our simulations for *x*(*t*) and *z*(*t*), we computed the filter time constants *σ*_{d}, *σ _{f}* using (35) and . This approach is not strictly necessary for the DA model since the sequences

*X*

_{n}and

*Z*

_{n}can be computed analytically as well as the filter time constants

*σ*

_{dep},

*σ*and

_{fac}*σ*

_{dep+fac}, which convey the same dynamic information as

*σ*

_{d},

*σ*and

_{f}*σ*

_{d}

_{+f}. The calculation of these time constants is possible since the filter sequences involved a single

*n*-dependent term. However, this is not the case for Δ

*S*and Δ

_{dep,n}*S*, which are linear combinations of

_{fac,n}*n*-dependent terms. On the other hand, the shapes of Δ

*S*and Δ

_{dep,n}*S*suggest these filters can be captured by the descriptive model by computing the appropriate parameters using the results of our simulations. We use the notation

_{fac,n}*F*

_{dep},

*F*

_{fac}and

*H*

^{×}(

*t*) =

*F*

_{dep}(

*t*)

*F*

_{fac}(

*t*). The solid lines in Figs. 11-A1 to -A3 and 12-A1 to A3 confirm this. In particular, parameter values can be found so that

*F*

_{dep}(

*t*),

*F*

_{fac}(

*t*) and

*H*

^{×}(

*t*) provide a very good approximation to Δ

*S*, Δ

_{dep,n}*S*and , respectively (gray solid lines).

_{fac,n}Using the descriptive model we computed the time constants and . Figs. 11-B and 12-B (blue) show the dependence of these time constants with the single-event depression and facilitation time constant *τ _{dep,}*

_{2}(=

*τ*

_{fac,}_{2}) for two values of

*τ*

_{dep,}_{1}(=

*τ*

_{fac,}_{1}) and Δ

_{spk}= 10 (

*f*

_{spk}= 100). These results capture the generic model behavior via rescalings of the type (29). A salient feature is the non-monotonicity of the curves for and (blue) in contrast to the mofotnhoetocunricvietys for

*σ*

_{d,}_{2},

*σ*

_{f,}_{2}and

*σ*

_{d}

_{+f,2}for the depression and facilitation second component (red).

#### 3.7.4 Degeneracy

The fact that the same type of descriptive envelope models such as the one we use here capture the dynamics of the temporal filters for both single and multiple depression and facilitation processes show it will be difficult to distinguish between them on the basis of data on temporal filters. In other words, the type of temporal filters generated by the DA model (single depression and facilitation processes) are consistent with the presence of multiple depression and facilitation processes interacting as described by the cross (distribute) model.

### 3.8 Persistence and disruption of temporal filters for Δ*S*_{n} = *X*_{n}Z_{n} in response to variable presynaptic spike trains

_{n}Z

_{n}

By design, the temporal filters discussed above emerge in response to periodic presynaptic spike trains with period Δ_{spk}. Here we consider more realistic, irregular presynaptic spike trains with variable (*n*-dependent) ISIs defined by the sequence . The natural candidates are Poisson spike trains (the ISI distribution follows a Poisson process with the parameter *r* representing the mean firing rate) [59, 75]. For Poisson spike trains both the mean ISI (< *ISI* >) and the standard deviation (SD) are equal to *r*^{−1} and therefore the coefficient of variation CV = 1. For Poisson spike trains with absolute refractoriness *ISI*_{min} , < *ISI* >= *r*^{−1} + *ISI* and CV = 1 − *ISI*_{min} < ISI >^{−1} [75], making them more regular. We use here *ISI*_{min} = 1 so that the irregularity remains high.

The presynaptic input spike-train variability causes variability in the output peak sequences *X*_{n}, *Z _{n}* and the synaptic update Δ

*S*

_{n}=

*X*.

_{n}Z_{n}#### 3.8.1 Perturbations to periodic presynaptic spike train inputs

To introduce some ideas, we consider perturbations of periodic presynaptic spiking patterns of the form
where Δ_{spk} is constant (*n*-independent) and is a sequence of real numbers. The exponential factors in (7)–(8) and (26) read
where *τ*_{stp} represents *τ*_{dep} or *τ*_{fac}.

If we further assume |*δ _{spk,n}/τ_{stp}*| ≪ 1 for all

*n*, then

Eqs. (7)–(8) have the general form
for *n* = 1, 2, … , *N*_{spk} − 1 with
and

Substitution of (61) into these expressions yields and

The last terms in these expressions are the corrections to the corresponding param-eters for the constant values of ∆_{spk} (*δ _{p}* = 0, remaining terms) and contribute to the variance of the corresponding expressions. One important observation is that these variances monotonically increase with decreasing values of ∆

_{spk}(increasing values of

*f*

_{spk}). A second important observation is the competing effects exerted by the parameters

*τ*

_{stp}(

*τ*

_{dep}and

*τ*

_{fac}) on the variance through the quotients

As *τ*_{stp} decreases (increases), *e*^{−Δspk/τstp} decreases (increases) and 1*/τ _{stp}* increases (decreases). In the limit,

*τ*

_{stp}→ 0,

*e*

^{−Δspk/τstp}→ 0 and 1

*/τ*→ ∞ and vice versa. Therefore, one expects the variability to change non-monotonically with

_{stp}*τ*

_{dep}and

*τ*

_{fac}.

To proceed further, we use the notation

Substituting into (87) in the Appendix A.2 we obtain
and
where *α _{X}* =

*Q*(

*a*

_{d},

*τ*

_{dep}) and

*α*=

_{Z}*Q*(

*a*). In (73) and (74) the first two terms correspond to the solution (24) and (25) to the corresponding systems in response to a presynaptic spike train input with a constant ISI Δ

_{f}, τ_{fac}_{spk}(

*δ*= 0), which were analyzed in the previous sections. The remaining terms capture the (first order approximation) effects of the perturbations

_{p}*δ*= {

_{p}*δ*}, which depend on the model parameters through

_{spk,n}*α*,

_{X}*β*,

_{X}*α*,

_{Z}*β*and the sequences

_{Z}*δ*,

_{α,X,n}*δ*,

_{β,X,n}*δ*and

_{α,Z,n}*δ*defined by the equations above.

_{β,Z,n}These effects are accumulated as *n* increases as indicated by the sums. However, as *n* increases, both *Q*(*a*_{d}, *τ*_{dep})^{n} and *Q*(*a _{f} , τ_{fac}*)

^{n}decrease (they approach zero as

*n*→ ∞) and therefore the effect of some terms will not be felt for large values of

*n*provided the corresponding infinite sums converge. On the other hand, the effect of the perturbations will be present as

*n*in other sums. For example, for

*k*=

*n*– 1 in the last sums in (73) and (74), and therefore both

*δβ,X,n*−1 and

*δ*

_{β,Z,n}_{−1}will contribute

*X*

_{n}and

*Z*

_{n}, respectively, for all values of

*n*. For small enough values of

*δ*, the response sequences

_{p}*X*

_{n}(

*δ*) and

_{p}*Z*

_{n}(

*δ*) will remain close

_{p}*X*

_{n}(0) and

*Z*

_{n}(0), respectively (the response sequences to the corresponding unperturbed, periodic spike train inputs) and therefore the temporal filters will persist. Fig. 13 shows that this is also true for higher values of

*δ*. There, the sequence

_{p}*δ*was normally distributed with zero mean and variance

_{p}*D*= 1. In all cases, the mean sequence values computed after the temporal filter decayed (by taking the second half of the sequence points for a total time

*T*

_{max}= 100000,

*X*

_{c}and

*Z*

_{c}) coincides to a good degree of approximation with fixed-point of the unperturbed sequences and (compared the corresponding solid and dotted curves).

These results also confirm, by inspection, the previous theoretical observations. First, the variability is smaller for *τ*_{dep} = *τ*_{fac} = 500 than for *τ*_{dep} = *τ*_{fac} = 100. Second, as Δ_{spk} decreases *f*_{spk} increases), the variability increases. For *τ*_{dep} = *τ*_{fac} = 100 (Fig. 13-A) the variability of Δ*S*_{c} = *X _{c}Z_{c}* is smaller than the variabilities of both

*X*

_{c}and

*Z*

_{c}. For

*τ*

_{dep}=

*τ*

_{fac}= 100 (Fig. 13-B), the variability of Δ

*S*

_{c}=

*X*is smaller than the variability of

_{c}Z_{c}*X*

_{c}, but not always smaller than the variability of

*Z*

_{c}.

While this approach is useful to understand certain aspects of the temporal synaptic update filtering properties in response to non-periodic presynaptic spike train inputs, it is limited since it does not admit arbitrarily large perturbations, which could cause the perturbed ISI to be negative. One solution is to make phase-based perturbations instead of time-based perturbations. But it is not clear whether comparison among patterns corresponding to different Δ_{spk} are meaningful.

#### 3.8.2 Poisson distributed presynaptic spike train inputs

Fig. 14 shows the initial response of the depression (*X*_{n}), facilitation (*Z*_{n}) and synaptic update peak (Δ*S*_{n}) sequences to Poisson distributed presynaptic spike train inputs for representative values of the spiking mean rate *r*_{spk} and the depression and facilitation time constants *τ*_{dep} and *τ*_{fac}. A comparison between these responses and the temporal filters in response to periodic presynaptic spike inputs with a frequency equal to *r*_{spk} qualitatively shows that the temporal filtering properties tend to persist with different levels of fidelity. In Figs. 14-B2 and -C2, the temporal band pass-filter is terminated earlier than the corresponding deterministic one. In Fig. 14-B1, the temporal band-pass filter is initiated earlier than the corresponding deterministic one.

In Fig. 15 we briefly analyze the response variability of the *X*_{n}, *Z*_{n} and Δ*S*_{n} sequences induced by the presynaptic ISI variability. For *X*_{n} and *Z*_{n}, the variability decreases with increasing values of *τ _{dep}* and

*τ*

_{fac}in an

*r*

_{spk}-dependent manner (Fig. 15-A). For most cases, the variability also decreases with increasing values of

*r*

_{spk}in a

*τ*

_{dep}- and

*τ*

_{fac}-manner (Fig. 15-B). An exception to this rule is shown in Fig. 15-B1 for the lower values of

*r*

_{spk}.

The variability of the Δ*S*_{n} sequences is more complex. Figs. 15-A1 and -A2 show examples of the peaking at intermediate values of *τ*_{dep} and *τ*_{fac}. Fig. 15-B1 shows an example of the variability reaching a trough for an intermediate value of *r*_{spk} and relatively low values of *τ*_{dep} and *τ*_{fac}, while Figs. 15-B2 and -B3 show examples of the variability peaking at intermediate values of *r*_{spk} for higher values of *τ*_{dep} and *τ*_{fac}. This different dependence of the variability with the model parameters and input rates emerge as the result of the different variability properties of the product of the sequences *X*_{n} and *Z*_{n}. A more detailed understanding of these properties is beyond the scope of this paper.

### 3.9 The MT model exhibits similar filtering properties as the DA model: Dynamics of the depression (*R*) and facilitation (*u*) variables and their interaction

The main difference between the DA model (5)–(6) and the MT model (12)–(13) is the update of the depression variables (*x* in the DA model and *R* in the MT model). Notation aside, the dynamics for depression and facilitation variables *x* and *z* in the DA model are completely independent both during the presynaptic ISI and the update. In the MT model, in contrast, while the dynamics of the depression and facilitation variables *R* and *u* are independent during the presynaptic ISI as well as the *u*-update, the *R*-update is dependent on *u*^{+}. As a result, the difference equations describing the peak sequence dynamics for *R*_{n} and *u*_{n} (14)–(15) (peak envelope responses to periodic presynaptic inputs) are not fully independent, but the equation for *R*_{n} is forced by the sequence *u*_{n}, which is independent of *R*_{n}. For the DA model, the difference equations for *Z*_{n} and *Z*_{n} are independent (7)–(8). Naturally, the steady-states ( and ) in the DA model are independent, while in the MT model, the steady state depends on the steady state . Here we show that despite these differences and the increased difficulty in the interpre-tation of the analytical solution for the MT model as compared to the DA model for the determination of the long-term depression and facilitation filter time constants, the two models describe similar dynamics.

Standard methods (see Appendix A) applied to these linear difference equations for Δ_{spk,n} = Δ_{spk} (independent of *n*) yield
and
where *Q*(*U*, *τ*_{fac}) is given by (26).

Because of the complexity of (75) we are not able to use the approach described in Section 3.3.1 to compute the (long-term) history-dependent time scales *σ*_{dep} and *σ _{fac}* in terms of the single event time scales (

*τ*

_{dep}and

*τ*

_{fac}) given by eq. (28). Instead, we use the descriptive modeling approach described in Section (3.3.2) by eqs. (32)–(35) and Section 3.3.3.

## 4 Discussion

In this paper, we set out to understand the mechanisms of generation of low-, high- and band-pass temporal filters in response to presynaptic spike trains in the presence of short-term plasticity in a feedforward network consisting of a periodic presynaptic spike train input synaptically connected to a postsynaptic cell in the presence of STP. We carried out a systematic analysis of the dependence of long-term time scales of the emergent temporal filters in terms of the time scales of the participating network components (the depression and facilitation time constants *τ*_{dep}, *τ*_{fac}, the synaptic decay time constant *τ*_{dec}, the membrane time constant *τ*_{m} and the presynaptic ISI) using two biophysically plausible phenomenological models that have been widely used in the literature [21, 40, 57–65]. In the DA model [59], the depression and facilitation variables evolve independently and their interaction occurs at the level of the synaptic update Δ*S*. In the MT model [58], the evolution of the depression variable is affected by the facilitation variable, in addition for both to interact at the synaptic update level. While, as we showed, both are analytically solvable, the advantage of the simpler DA model is that one can extract the temporal filter time scales from the analytical solutions. For the MT model, we have developed the descriptive model developed for that purpose (see Supplementary Material Section for the more detailed analysis). This descriptive modeling approach can be used and extended to more complex models for which analytical solutions are not available. Our results make a number of predictions that can be experimentally tested both *in vitro* and *in vivo* using dynamic clamp [76–78] and optogenetics [79, 80].

An important question we addressed in our work is how the single event time constants (e.g., *τ*_{dep}, *τ*_{fac}, *τ*_{dec}), which control the systems’ dynamics during the ISIs, are communicated to the temporal filters. In other words, how the temporal filters’ long-term time constants (*σ*_{dep}, *σ _{fac}*,

*σ*

_{sum}) depend on the single-event time constants. For the simple synaptic model with one-dimensional linear dynamics for the variable synaptic variable

*S*and a constant update Δ

*S*(independent of

*S*), the single-event and temporal filter time constant coincide. For the slightly more complex models for depression and facilitation, which have also linear one-dimensional dynamics during the ISI, but the updates depend on the depression and facilitation variables, there is a departure of the temporal filter time constant from the single event time constant. The dependence between the two is more complex and involves the corresponding update parameters as we showed analytically for the DA model and using the descriptive approach for the MT model. This reflects the dependence of the stationary properties of the temporal filters with the single event parameters.

Temporal band-pass filters are expected out of the multiplicative interaction between a low- and a high-pass filter under certain conditions, as we illustrated by using the descriptive model consisting of exponential functions decaying towards prescribed constant values. Similar to the depression and facilitation temporal filters, which are generated by one-dimensional linear difference equations, the band-pass filters can be considered as an overshoot type of solution to a linear difference equation. Overshoots are not possible for one-dimensional linear difference equations, but they are for two-dimensional linear difference equations, implying that two time scales would be enough to explain the properties of the band-pass filter. However, our results indicate that a third time scale, combining the first two, is needed to explain the band-pass filter properties.

The interaction between the depression/facilitation temporal filters with the synaptic and postsynaptic dynamics via de synaptic update generates additional low-, high- and band-pass temporal filters with additional time scales. For relatively low membrane time constants, the postsynaptic dynamics reflected the synaptic dynamics, but for higher membrane time constants, the postsynaptic filters departed from the synaptic ones, exhibiting temporal band-pass filters in the presence of synaptic temporal low-pass filters or in the presence of synaptic temporal band-pass filters, but peaking at different times.

In order to account for more realistic situations, we considered scenarios where more than one depression and facilitation processes with different time constants interact and scenarios where the presynaptic inputs are no longer periodic, but exhibit different types of variability including bursting and Poisson-distributed spike trains. The results are consistent with the previous ones in the paper. However, more research is needed to investigate these scenarios in more detail and, in particular, to determine whether and under what circumstances the responses to variable input spiking patterns can be understood in terms of the results for periodic input spiking patterns.

While biophysically plausible, the phenomenological models of STP we used in this paper are relatively simple and leave out a number of important biological details that might contribute to determining the properties of STP-dependent temporal filters and their consequences for information processes. Further research is needed to understand the properties of these filters and how they emerge as the result of the interaction of the building blocks. An additional aspect that requires attention is the possible effect of astrocyte regulation of STP [10, 11] on the mechanisms of generation of STP-dependent temporal filters. Our work leaves out the mechanisms of generation and properties of the stationary low-, high- and band-pass filters and the associated phenomenon of synaptic and postsynaptic resonance. This will be discussed elsewhere.

The conceptual framework we developed in this paper allows developing ideas on the mechanisms of generation and properties of postsynaptic temporal filters in response to presynaptic inputs in the presence of STP. An important aspect of this framework is the separation of the feedforward network into a number of building blocks, each one with its own dynamics. The emerging temporal filters can be analyzed in terms of the hierarchical interaction of these building blocks. This conceptual framework can be used to investigate the properties of low-, high- and band-pass stationary, frequency-dependent filters and the emergence of synaptic and postsynaptic resonances, and it can be extended to include a number of more complex scenarios, including non-periodic synaptic spike trains (e.g., Poisson spike inputs, bursting patterns with two or more spiking frequencies), more complex networks (e.g., two recurrently connected cells with STP in both synapses, three-cell feedforward networks with STP in both synapses), the modulatory effects of astrocytes, more complex postsynaptic dynamics involving ionic currents that have been shown to produce resonances [81–86], and the generation of postsynaptic spiking temporal filters.

## A 1D linear difference equations

### A.1 Constant coefficients

Consider the following linear difference equation
where *α* and *β* are constants. The steady-state for this equation, if it exists, is given by

By solving (77) recurrently and using
where *a* ≠ 1 is a real number, one gets

Substitution of (78) into this equation yields

Application of formula (81) to the difference equations (7) and (8) gives, respectively, and

### A.2 Variable (*n*-dependent) coefficients

Consider the following linear difference equation

By solving (77) recurrently one gets where we are using the convention . Eq. (85) reduces to eq. (83) if both coefficients in (85) are constant.

Consider now eq. (84) where the coefficients are expressed as small perturbations *δ _{α,n}* ≪ 1 and

*δ*≪ 1 (

_{β,n}*n*= 1, 2, …), respectively, of constant coefficients

To the first order approximation, the solution (85) reads

## B Some properties of and and their dependence with Δ_{spk} and *τ*_{dep/fac}

Consider and given by (9) and (10), respectively.

### B.1 Monotonic dependence of and with Δ_{spk}

If *a*_{d} > 0 and *x*_{∞} > 0, then then is an increasing function of Δ_{spk} and a decreasing function of *f*_{spk} This results from
if *a*_{f} < 1 and *z*_{∞} < 1, then is a decreasing function of Δ_{spk} and an increasing function of *f*_{spk}. This results from

### B.2 Monotonic dependence of and with *τ*_{dep/fac}

If *a _{d}* > 0 and

*x*

_{∞}> 0, then is an decreasing function of

*τ*

_{dep}. This results from

If *a*_{f} < 1 and *z*_{∞} < 1, then is a decreasing function of *τ*_{fac}. This results from

## C Models of synaptic depression and facilitation

### C.1 Depression - facilitation model used in [74]

Following [72, 73], the synaptic variables *S* obey a kinetic equation of the form
where *N* (*V*) (mM) representes the neurotransmitter concentration in the synaptic cleft. Neurotransmitters are assumed to be released quickly upon the arrival of a presynaptic spike and remain in the synaptic cleft for the duration of the spike (~ 1 ms). This can be modeled by either using a sigmoid function
or a step function if the release is assumed to be instantaneous. The parameters *τ*_{r} and *τ*_{d} are the rise and decay time constants respectively (msec).

This model assumes *N* (*V*) is independent of the spiking history (the value of *N* (*V*) during a spike is constant, except possibly for the dependence on *V*). (There is evidence that this is not realistic [70, 87].) In [74], the “activated” time was 1 ms [88, 89].

In [74], the follow the description of the synaptic short-term dynamics following [40, 57]. They use [*T*] instead of *N* (*V*) where [*T*] is constant during the release time and zero otherwise, and they define [*T*] = *κ* Δ*S*_{n}. Effectively,

### C.2 Depression model used in [90]

Following experimental procedures described in [91], the synaptic current is described by *I*_{syn} = *G*_{ex}*ad*(*V* − *E*_{ex}) where *a* and *d* are variables that represent activation and depression processes, respectively. They follow the form:
where *y* = *a, d*. The steady-state of *y* is given by
and its time constant follows

This model is used in [90] to describe bistability in pacemaker networks with recurrent inhibition and depressing synapses. Parameters in these equations are experimentally fitted from the pyloric network of the crab *Cancer borealis*.

## D Additional model formulations for multiple depression-facilitation processes

In Section 3.7 we discussed the model formulation (51)–(52) describing the interplay of two depression-facilitations processes. A number of additional, simplified formulations are possible based on different assumptions. The models we propose here are natural mathematical extensions of the single depression/facilitation processes discussed in the main body of this paper. They are phenomenological models, not based on any experimental observation or theoretical foundation, and they are limited in their general applicability. However, they are useful to explore the possible scenarios underlying the interplay of multiple depression and facilitation time scales affecting the PSP dynamics of a cell in response to presynaptic input trains.

### D.1 Additive and multiplicative segregated-processes models

In the additive and multiplicative *segregated* models, the variable *M* is given, respectively, by
and
where the parameter *α* ∈ [0, 1] controls the relative contribution of each of the processes. Correspondingly, the updates are given by
and

For *α* = 0, and reduce to Δ*S*_{1,n} (single depression-facilitation process). This accounts for the regimes where *τ*_{dep,2}, *τ*_{fac,2} ≪ 1. If the two processes are equal (*τ*_{dep,1} = *τ*_{dep,2} and *τ*_{fac,1} = *τ*_{fac,2}), then and also reduce Δto*S*_{1,n}. However, these models fail to account for the reducibility in the situations where only *τ*_{dep,2} ≪ 1 or *τ*_{fac,2} ≪ 1, but not both. The option of considering depression to be described by *x*_{1} and facilitation by *z*_{2} (with *τ*_{fac,1}, *τ*_{dep,2} ≪ 1) is technically possible in the context of the model, but it wouldn’t be consistent with the model description of single depression-facilitation processes, and it will make no sense to use the model in this way. In general, this model would be useful when the depression and facilitation time scales for each process 1 and 2 are comparable and the differences in these time scales across depression/facilitation processes should be large enough.

### D.2 Fully multiplicative model

One natural way to extend the variable *M* to more than one process is by considering
and the synaptic update, given by

This formulation presents us with a number of consistency problems related to the reducibility (or lack of thereoff) to a single depression-facilitation process in some limiting cases when, for example, the two depression or facilitation time constants are very similar and therefore the associated processes are almost identical, or the depression or facilitation time constants are very small and therefore the envelopes of the associated processes are almost constant across cycles.

More specifically, first, if *τ*_{dep,2}, *τ*_{fac,2} ≪ 1 (almost no STD), then for all *n* after a very short transient and therefore . One way, perhaps the simplest, to address this is to divide the expressions (102) and (103) by and redefine Δ*S*_{k,n} for the single depression-facilitation process accordingly. Specifically,
where we use the notation

The effect of redefining Δ*S*_{k,n} by dividing the original expression (used in the previous sections) does not affect the time constants and the differences in the values between the two formulations is absorbed by the maximal synaptic conductance.

Second, if *τ*_{dep,1} = *τ*_{dep,2} and *τ*_{fac,1} = *τ*_{fac,2}, then *X*_{1,n} = *X*_{2,n} and *Z*_{1,n} = *Z*_{2,n} for all *n*, and instead of . In order to address this, the synaptic update can be modified to
where
and *H*(Δ*τ*) is a rapidly decreasing function satisfying *H*(0) = 2 and lim_{Δτ→∞} *H*(Δ*τ*) = 1. In our simulations we will use
with *β* > 0. Correspondingly,

In this way,

If

*τ*_{dep,1}=*τ*_{dep,2}, then*X*_{1,n}=*X*_{2,n}for all*n*and*λ*_{dep}= 1/2. This givesIf, in addition,

*τ*_{fac,1}≠*τ*_{fac,2}and |*τ*_{fac,1}−*τ*_{fac,2}| > 0 is large enough, then*λ*_{fac}= 1 andIf

*τ*_{fac,1}=*τ*_{fac,2}, then*Z*_{1,n}=*Z*_{2,n}for all*n*,*λ*_{fac}= 2 andIf, in addition,

*τ*_{dep,1}≠*τ*_{dep,2}and |*τ*_{dep,}_{1}−*τ*_{dep,2}| > 0 is large enough, then*λ*= 1 and_{dep}It follows that if both

*τ*_{dep,1}=*τ*_{dep,2}and*τ*_{fac,}_{1}=*τ*_{fac,}_{2}, then*X*_{1,n}=*X*_{2,n}and*Z*_{1,n}=*Z*_{2,n}for all*n*,*λ*_{dep}=*λ*_{fac}= 2 andIf

*τ*_{dep,2}≪ 1 and |*τ*_{dep,1}−*τ*_{dep,2}| is large enough, then*X*_{2,n}= 1 for all*n*(after a very short transient),*λ*_{dep}= 1, and thenIf, in addition,

*τ*_{dep,1}≤ 1 and*τ*_{dep,2}~*τ*_{dep,1}(|*τ*_{dep,1}−*τ*_{dep,2}| ~ 0 not large enough), then*X*_{1,n}= 1 for all*n*(after a very short transient),*λ*_{dep}= 2, and thenIf

*τ*_{fac,2}≪ 1 and |*τ*_{fac,1}−*τ*_{fac,2}| is large enough, then*Z*_{2,n}=*a*_{f}for all*n*(after a very short transient),*λ*_{fac}= 1, and thenIf, in addition,

*τ*_{fac,1}≪ 1 and*τ*_{fac,2}~*τ*_{face,1}(|*τ*_{fac,1}−*τ*_{fac,2}| ~ 0 not large enough), then*Z*_{1,n}=*a*_{f}for all*n*(after a very short transient),*λ*_{fac}= 2, and thenIt follows that if

*τ*_{dep,1},*τ*_{dep,2}≪ 1 ( |*τ*_{dep,1}−*τ*_{dep,2}| ~ 0 not large enough) and*τ*_{fac,1},*τ*_{fac,2}≪ 1 (|*τ*_{fac,1}−*τ*_{fac,2}| ~ 0 not large enough), then

## Supplementary Material

### The Markran-Tsodyks (MT) model

First, we briefly remark on the notation used in this section. We conduct analysis using continuous extensions of Δ*S*_{n}, *S*_{n}, and *V*_{n} – denoted as Δ*S*, *S*, *V* in the forthcoming figures and text. Time scales of temporal HPFs and LPFs in *S* are denoted *σ*_{f,S} and *σ _{d,S}*. The third time scale in temporal BPFs in

*S*are denoted

*σ*

_{d+f,S}. All temporal filters are fitted using gradient descent of a quadratic cost function.

The analysis for the MT model proceeds similarly to the DA models’. As seen in Section 3.3, the interaction between *X* and *Z* produces low-, high-, and band-pass temporal filters in Δ*S*. Similarly, *R* and *u* produce low-, high-, and band-pass temporal filters in Δ*S*. Again, we find Δ*S* temporal LPFs and HPFs not only develop in synapses exhibiting exclusively STD and STF, respectively. Indeed, Δ*S* temporal LPFs (HPFs) can develop in synapses where the time scale of depression (facilitation) dominates facilitation (depression). However, as in the case of the DA model, the exact ranges of *f*_{spk} over which LPFs and HPFs develop depend on the balance between facilitation and depression. Figure S1-A1 shows that almost exclusively depressive synapses exhibit LPFs for 0 < *f*_{spk} < 150, whereas a dominantly depressive synapse may stop producing LPFs for *f*_{spk} > 100 (compare to Figure S1-A5). A similar situation arises in facilitating synapses (Figure S1-A2 and -A5).

*R*_{n} and *u*_{n} are well described by exponential decays, much in the same way *X*_{n} and *Z*_{n} are observed to be (Section 3.4). As such, as we did in the DA model, in the MT model we imagine that temporal filters in Δ*S* are heuristically the product of two exponentials. Despite the non-linearity present in MT model which complicates the analysis of how the long-term time scales of *R*_{n} and *u*_{n} are passed through their product Δ*S*, Δ*S* temporal LPFs (examples in Figures S2-B2,-B3) and HPFs (examples in Figures S2-A2,-A3) are still well described by a single time scale exponential. The time scales extracted from HPFs at dominantly facilitating and exclusively facilitating synapses are summarized in Figure S2-A1. Figure S2-B shows analogous results for LPFs of the MT model. A careful reader will note that Figure S2 refers to temporal filters of *S*, rather than Δ*S*. However, *τ*_{dec} = 3 for these figures so that the contribution of the synaptic HPF implemented by synaptic decay is inconsequential for this discussion. The same remark also applies to Figures S3, S6, and S7.

In Section 3.6, BPFs in the DA model are shown to arise from 3 time scales, 2 of which can be extracted from corresponding LPFs and HPFs. A similar result is true for the MT model. Figure S3 outlines how these three time scales vary as the input frequency varies. Band-pass temporal filters become more sharply peaked as the input frequency increases. This reflects itself as observable decreases in the scenario where three time scales characterize band-pass temporal filters. Figure S3-B show that this third time scale is not superfluous – that removing the *σ*_{d+f,S} time scale from the model drastically alters the temporal filter fit.

Finally, we briefly review how Δ*S* temporal LPFs, HPFs, and BPFs are propagated to *S* and PSP. We note that *τ*_{dec} and *τ*_{mem} implement temporal HPFs. Figure S4 summarizes the types of filters that result in *S* from different incident Δ*S* temporal filters. Similarly, Figure S5 summarizes the types of filters that result in *V* from different incident *S* temporal filters. As in the DA model, we note there are instances where BPFs are passed through levels of organization and others where they arise due to an interaction of filters at different levels of organization (see Section 3.7). The communication through BPFs between levels of organization is exemplified by Figure S4-A3 (Δ*S* to *S*) and Figure S5-A3 (*S* to *V*). BPFs arising from interactions of filters between levels of organization are exemplified by Figure S4-A1 (Δ*S* to *S*) and Figure S5-A1 (*S* to *V*).

BPFs in the PSP arise in two ways – either passed through from the incident BPF filter or as the result of interacting an incident LPF and HPF implemented by the post-synaptic cell. These BPFs can be distinguished by analyzing the three time constants used to fit BPFs. First we consider synaptic BPFs that transfer to the post-syntactic cell’s response.

In this case, the procedure to extract the three time constants of the PSP BPF differs slightly from the procedure used to extract the three time constants of BPFs in *S* and Δ*S*. Instead of fixing LPF and HPF time scales and finding the best fitting third time constant, as was done in the Δ*S* BPFs, all three time constants for the PSP BPF are allowed to vary. In this way, we find a triple of time constants that fit the PSP BPF: *ρ _{a}, ρ_{b}* and

*ρ*(Example shown in Figure S6-A3). Then we introduce the following quantities:

_{c}These time constants are then compared to the magnitude of the time constants obtained from the synaptic BPF. In Figure S6-C we note that the synaptic BPF has time constants such that *σ*_{d,S} > *σ*_{f,S} > *σ*_{d+f,S}. Assuming the post-synaptic cell maintains this relation, we associate *ρ*_{1} with *σ*_{d,S}, *ρ*_{2} with *σ*_{f,S} , and *ρ*_{3} with *σ*_{d+f,S}. Figure S6-C suggests that the HPF implemented by the membrane time constant is modifying all three time constants of the incident synaptic BPF.

BPFs in the post-synaptic cell are implemented also by the interaction with synaptic LPFs and the HPF implemented by the membrane time constant. One may use the same three parameter model to fit these PSP BPFs. However, we find that there are actually two time scales describing the shape of these BPFs. Figure S7-B show explicit examples of the third time scale’s insignificance. Furthermore, one of the time scales describing the PSP BPF is shown to be inherited from the incident LPF’s time scale of decay.

Figure -B1 and -B2 show how the membrane time constant also modifies incident LPF and HPF time constants in the post-synaptic cell. Here, the same models to extract time constants of rise for HPFs and time constants of decay for LPFs in Δ*S* also work well for extracting time constants for LPFs and HPFs of the PSP (example fits in Figure S6-A1 and -A2). In this figure, *σ*_{f,V} is the time constant of rising in PSP HPFs and *σ*_{d,V} is the time constant of decay in PSP LPFs, respectively.

In Section 3.9.2, we discuss how the temporal filters implemented by *Z*_{n}, *X*_{n}, and Δ*S*_{n} persist in the presence of Poisson spiking. Here we review a potential consequence of Poisson inputs interacting with temporal filters: gain control. The data in the following figure, Figure S8, uses the MT model, however, the foregoing discussion suggests that the DA and MT model both exhibit temporal filters, albeit via slightly differing quantitative mechanisms. Figure S8 plots average amplitude of voltage response over 1100 trials where the input Poisson rates change over time. As the spiking rates change over time, different steady states are achieved. Overshooting transient features develop between rate changes when STD time scales increase. The fact that these overshoots depend on the time scale of STD suggests that there may be a connection between the overshoot magnitudes and the LPFs that STD implement. Furthermore, the magnitude of these average rate changes and their dependence on the initial and final rates was studied as a mechanism for gain control by Abbott *et. al.*. Figure S8 show that average amplitude also depends on the time scale of STD and membrane time scales - ergo, it follows that filters that STD and membrane time constants implement may also play an important role in determining the average amplitude of voltage response.

## Acknowledgments

This work was partially supported by the National Science Foundation grant DMS-1608077 (HGR) and an NSF Graduate Research Fellowship (YM). The authors are grateful to Allen Tannenbaum for useful comments and support, and to Farzan Nadim, Dirk Bucher and Nelly Daur for useful discussions.

## Footnotes

↵‡ Graduate Faculty, Behavioral Neurosciences Program, Rutgers University; Corresponding Investigator, CONICET, Argentina.

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