An oscillatory mechanism for multi-level storage in short-term memory

Oscillatory activity is commonly observed during the maintenance of information in short-term memory, but its role remains unclear. Non-oscillatory models of short-term memory storage are able to encode stimulus identity through their spatial patterns of activity, but are typically limited to either an all-or-none representation of stimulus amplitude or exhibit a biologically implausible exact-tuning condition. Here we demonstrate a simple mechanism by which oscillatory input enables a circuit to generate persistent or sequential activity that encodes information not only in the spatial pattern of activity, but also in the amplitude of activity. This is accomplished through a phase-locking phenomenon that permits many different amplitudes of persistent activity to be stored without requiring exact tuning of model parameters. Altogether, this work proposes a class of models for the storage of information in working memory, a potential role for brain oscillations, and a dynamical mechanism for maintaining multi-stable neural representations.


Abstract
Oscillatory activity is commonly observed during the maintenance of information in shortterm memory, but its role remains unclear. Non-oscillatory models of short-term memory storage are able to encode stimulus identity through their spatial patterns of activity, but are typically limited to either an all-or-none representation of stimulus amplitude or exhibit a biologically implausible exact-tuning condition. Here, we demonstrate a simple phaselocking mechanism by which oscillatory input enables a circuit to generate persistent or sequential activity patterns that encode information not only in their location but also in their discretely graded amplitudes.

Significance
A core observation in many memory systems and tasks is the presence of oscillations during memory maintenance. Here, we demonstrate a mechanism for the accumulation and storage of information in short-term memory in which oscillatory activity enables a solution to long-standing challenges in modeling the persistent neural activity underlying working memory. These challenges include the ability to encode information with low firing rates, multi-level storage of stimulus amplitude without extreme fine tuning, and multi-level storage of information in sequential activity. Altogether, this work proposes a new class of models for the storage of information in working memory, a new potential role for brain oscillations, and a novel dynamical mechanism for multi-stability.
The maintenance of information in short-term memory is a key component of a wide array of cognitive (1,2) and non-cognitive (3,4) functions. However, the biophysical mechanisms that enable memory storage over the seconds-long time scale remain unclear.
Single-unit studies have demonstrated a neural correlate of memory maintenance in the persistent activation of neurons whose population activity spans the memory period (reviewed in (2,5,6)). Theoretical studies have shown how such persistent activity can be generated by recurrent network feedback (7)(8)(9), but simple instantiations of this idea are either implausibly sensitive to mistuning or can only maintain a single elevated firing rate that is unrealistically high (the 'low firing rate problem', reviewed in (4,10)), limiting storage about a given item to a single bit ('on' or 'off') of information.
Separately, previous studies have identified distinct bands of oscillatory activity in field potential recordings and EEG during the maintenance of working memory (reviewed in (11)). Such activity can be generated through cell-intrinsic mechanisms, local circuitry, or longrange interactions (12)(13)(14). However, it remains an open question whether oscillatory activity is necessary, sufficient, or even beneficial for working memory storage. Previous work has proposed how oscillations can contribute to a variety of memory functions such as the generation or maintenance of persistent activity (15,16); the structuring of spatial codes through frequency coupling (17); and the = 0), changes in the firing activity ( ) of a population are determined by the relative balance of network feedback (black, ( )) and neuronal decay processes (gray, − ). (B,C) Nonlinear models typically exhibit a 'low firing rate problem'. (B) During the memory period when external input is absent, the intersections of the decay (gray) and network feedback (black) functions are such that there are no stable fixed points (solid circles) within the range of firing rates typically observed during persistent neural activity. (C) Firing rates below the unstable fixed point (B,C, open circle) decay to zero (green, purple, orange lines), while firing rates above the unstable fixed point run off to uncharacteristically high rates (red, blue lines). (D,E) Linear models exhibit the 'fine tuning problem': minute changes in the strength of feedback (red: +5%, orange: -5%) relative to the tuned value (blue) result in instability and the inability to maintain stable persistent activity. coordination, control, and gating of memory-related activity (18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29). By contrast, other studies have suggested that oscillations could be an epiphenomenon of other computational or network mechanisms (30)(31)(32). Here, we demonstrate a potential mechanistic role for oscillations, regardless of source or frequency, by showing how the addition of oscillatory inputs to simple recurrent feedback circuits can enable both low firing rate persistent activity and a discretely graded set of persistent firing rates that increases the information capacity of a memory network.
To illustrate the core challenges that arise when generating biologically plausible models of persistent activity, consider an idealized circuit consisting of a memory neuron (or lumped population) connected to itself through positive feedback (Fig. 1A); this basic motif of recurrent excitation is the key component of most circuit models of persistent neural activity (reviewed in (4)). This simple circuit receives a brief stimulus (Figs. 1A,C,E external input) and needs to store it through persistent activity. Stable persistent activity ( !" !# = 0, > 0) is achieved only when the intrinsic decay of the neuron (represented by the term − ) and the recurrent drive to the neuron ( ( )) are equal in magnitude and cancel each other. This condition imposes two separate, but related, problems that depend on whether the rate function ( ) is linear or nonlinear. In the typical nonlinear case (Fig.  1B,C), if the stimulus is too weak, the memory neuron's low initial firing rate provides insufficient recurrent feedback to overcome the post-stimulus intrinsic decay of activity (Fig. 1B, left of open circle). As a result, the firing rate of the network returns to a low (or zero) baseline firing rate (Fig. 1C, orange, purple, green traces). By contrast, if the stimulus is stronger, the memory neuron's initial firing provides recurrent feedback that exceeds the rate of intrinsic decay (Fig. 1B, right of open circle), leading to a reverberatory amplification of activity in which the rate rises until some saturation process brings the rate to rest at an elevated persistent level (Fig. 1C, blue and red traces). Thus, the only possibilities are that activity decays to its baseline level or that activity runs away to saturation at a high level of activity that, for typical neuronal nonlinearities, is unrealistically large. A different problem emerges in the case where the rate function ( ) is linear (Figs. 1D,E). The linearity of the rate function in this case allows a continuum of persistent rates, corresponding to the continuous set of points at which the feedback and decay lines overlap, to be stored (Fig. 1D, blue line), unlike the nonlinear case. This comes at the cost of a 'fine-tuning' condition: the strength of the recurrent synapse(s) must be exactly tuned to counterbalance the strength of the rate decay; an arbitrarily small violation of this condition causes the rate to exhibit runaway growth ( Fig. 1E red trace) or decay to a low baseline ( Fig. 1E orange trace). Although presented here for a very simple example, these problems are also commonly observed in larger neural networks (33).
We next illustrate what happens when a network with the same positive feedback architecture is provided with a subthreshold oscillatory drive (Fig. 2). We demonstrate this in the more biologically realistic case of a conductance-based spiking neuron model (34) that facilitates the phase-locking phenomenon that we will describe. Without an oscillatory input, the model exhibits the 'low firing-rate problem' (Figs. 2A,B) and can only maintain persistent activity at a high spiking rate or not spike at all. When a subthreshold oscillation is added to the model (Figs. 2C-F), the oscillatory drive has two effects. First, it provides extra input that allows small initial inputs to trigger low-rate spiking. Second, spiking of the memory neuron does not lead to runaway feedback because, before the feedback can run away, the oscillatory drive returns towards its trough, causing a cessation of spiking. The net result is that the spike-driven feedback becomes discretized, forming a  Figure 1B, the conductance-based spiking model exhibits stable fixed points only at zero and high firing rates (filled circles). Spiking rates between these two fixed points decay to zero from below the unstable fixed point (open circle) or run off to high rates from above the unstable fixed point. (C) Schematic of conductance-based neuron with the addition of an oscillatory baseline input. (D-F) Maintenance of discretely graded persistent activity levels enabled by baseline oscillatory input. Phase-locking to the oscillatory input creates stable fixed points at integer multiples of the baseline frequency. There is a trade-off between the number of firing rates that can be maintained and the robustness of these fixed points, which is related to the spacing between the fixed points. (D) Lower frequency oscillations enable a larger number of closely spaced fixed points. (E,F) Higher frequency oscillations lead to fewer, more robust, fixed points. staircase whose step heights correspond to the number of spikes emitted by the neuron per oscillation cycle (Figs. 2D-F). The phase locking of the spiking to the subthreshold oscillatory drive constrains these spike numbers to be integer multiples of the oscillation.
The key requirements for this mechanism to enable discretely graded persistent activity are the following: First, the oscillation must be strong enough to reset the activity at its troughs. Second, there must be some process that enables the activity from one cycle of the oscillation to carry through to the start of the next cycle and consequently enable renewed spiking as the oscillatory input heads towards its peak. For the simple case illustrated here, where all neurons receive oscillatory inputs that are perfectly aligned in phase, the mechanism enabling inter-cycle memory is a slow NMDA-like (or local dendritic) synaptic time constant (3,10,35). Alternatively, we show in the Supporting Information that, if there is heterogeneity in the phases of the oscillations received by the individual neurons in the network, the time between cycles may be bridged by the firing of other neurons in the network (Fig. S1). Some degree of tuning of the feedback is required to have multiple levels of response -such a tuning requirement is generic of models of analog or finely discretized persistent activity. In the present case, the width of the steps of the staircase provides a moderate level of robustness to mistuning, especially for higher oscillation frequencies (Figs. 2D-F). Mechanistically, this robustness occurs because errors in the tuning of feedback that are insufficient to systematically add or subtract an extra spike per cycle do not persist from cycle to cycle, unlike in models that have no oscillatory trough to reset (error-correct) the spiking activity. We illustrate Steady-state firing rates as a function of synaptic activity ( !"# in Equation 1, Methods) that is held at steady values; mistuning the autapse strength by +/-5% has no effect on the existence and location of the stable fixed points (intersections of black lines and horizontal stairs). (C) Responses of a traditional, approximately linear, conductance-based model of persistent neural activity (adapted from model of (37)). Detuning the recurrent feedback strength by 5% (orange and red traces) causes spiking activity to decay to 0 (orange, decreasing feedback strength) or run off to high rates (red, increasing feedback strength). (D) Small weight changes cause systematic loss of fixed points in the traditional model. this robustness to weight changes in Figure 3, where we compare the oscillatory autapse memory model (Fig. 3A,B) to an approximately linear autapse model (36,37) that can produce (nearly) graded persistent activity (Fig. 3C,D). Each model receives an arbitrary sequence of positive and negative input pulses, and must temporally accumulate and store the pulses in persistent activity. The linear spiking autapse model requires fine tuning to maintain persistent activity: very small deviations from the tuned autapse weight lead to activity that grows to a saturating level or decays to zero activity (Figs. 3C,D). In contrast, the same synaptic weight deviations have negligible effect on the accumulation and multi-level storage capability of the nonlinear spiking neuron with oscillatory drive (Figs. 3A,B).
The above examples demonstrate the basic mechanism by which oscillatory input may permit discretely graded levels of firing rate to be robustly stored in a recurrent excitatory network model of persistent activity. We next explored applications of this basic principle in the case of three different network architectures: a spatially uniform (all-to-all) network that temporally integrates its inputs (Fig. 4); a 'ring-like' architecture whose activity can store both a spatial location and discretely graded levels at that location (Fig.  5); and a chain-like architecture that can generate sequences of activity with multiple discretely graded amplitudes (Fig. 6).
We first extended the demonstration of temporal integration, shown in Figure 3, to a spatially homogeneous (all-to-all) network composed of 1000 neurons (Fig. 4A,B). This permitted us to not only examine the systematic mistuning of weights shown in the autapse network, which produces identical results in the averaged activity of the 1000 neuron network (Methods), but also to examine the robustness to four different sources of noise and variability: input noise, in which each neuron in the network received independent exponentially filtered noise added to the subthreshold oscillatory drive (Fig.  4C); noise in the connection weights, in which each synapse in the network was initialized with added random noise (Fig. 4D); randomly shuffled phases of the subthreshold oscillatory drive, in which each neuron received an oscillatory signal whose phase was randomly picked from a uniform distribution on [0, 2 ) at initialization (Fig. S2A); and noisy oscillation frequency and amplitude, in which the parameters of the subthreshold oscillatory drive underwent noisy drift (given by an Ornstein-Uhlenbeck process) during the simulations (Fig. S2B). In all of these cases, the network was able to accurately maintain multi-level persistent activity despite moderate perturbations. Figures 4C-D and Figure S2B illustrate the conditions for which the magnitude of the perturbations began to adversely affect network performance -for noise less than this amount, persistent activity was accurately maintained over a timescale of seconds, whereas larger noise levels led to progressively larger drifts of activity.
Next we demonstrate that a similar temporal integration of inputs can also occur in spatially structured networks. We consider a classic "ring model" architecture commonly used to model spatial working memory tasks in which stimuli can be presented at any of various locations arranged in a circular (ring-like) layout. The model consists of a ring of neurons with local excitatory connectivity and functionally wider inhibitory connectivity (Fig. 5A, Methods). Such models can generate persistent activity at any spatial location Although individual neurons in the network respond with different rates, the network is able to maintain persistent activity at many levels.
along the ring, but typically have only a binary "on-off" representation at a given spatial location (Fig. 5B). When we added an oscillatory input stimulus to such a ring model, the network could store multiple, discretely graded levels of activity at any spatial location (Fig. 5C) and could temporally integrate location-specific inputs into discretely graded levels (Fig. 5D). While the spatial memory (bump attractor) networks proposed in (38)(39)(40) are capable of generating graded persistent activity, the network presented here represents, to our knowledge, the first spatial memory network to encode multi-level activity without requiring an exact tuning condition.
Recent studies have shown that memory activity during a delay period also may take the form of a sequence of activity that spans the delay (41)(42)(43). Models of such activity typically generate chain-like patterns of activity that attain only a single, stereotyped level of firing rate. Consistent with this, when we constructed a network with a chain-like architecture (Fig. 6), we found that, in the absence of oscillatory input, the sequential network activity either quickly decayed when the initial stimulus amplitude was too small or converged to a single saturated level of activity for larger stimuli (Fig. 6B). By contrast, in the presence of a subthreshold oscillatory input, the network could exhibit sequential activity with discretely graded amplitudes for the same pattern of input (Fig. 6C). Thus, as in the persistently active networks, the oscillatory sequential memory network could encode multiple discretized stimulus levels.
In summary, this work demonstrates a simple mechanism by which oscillatory input to a memory network can transform it from storing only binary amplitudes to maintaining discretely graded amplitudes of persistent activity. Memory networks using this mechanism require a cellular, synaptic, or network process that can span the period of the oscillation, suggesting a possible tradeoff in memory storage: higher frequency oscillations do not require long timescale processes to span the oscillation cycle, but due to their short period may only store one or a few values ( Figure 2F); lower frequency oscillations could store more items, but require a process with longer timescale to bridge the troughs occurring in each cycle. Our work complements traditional attractor models of working memory that typically fall into two classes: bistable models that robustly maintain two levels of activity ( Figures 1B and 2B) and continuous attractor models that can maintain nearly analog storage of memory but require very precise tuning of connection weights (Figures 1D and 3C). Our model represents an intermediate possibility with relatively moderate tuning requirements (Fig.  3B) and a discretely graded set of response levels. Previous work (44,45) has suggested how multiple, spatially distinct bistable processes in a cell can be coupled together to form multiple stable levels of firing activity; here we demonstrate a complementary mechanism for forming multi-stable representations that relies on temporal, rather than spatial, patterning of inputs. Altogether, this work suggests a potential mechanism by which oscillatory activity, which is commonly observed during working memory tasks, may expand short-term memory capacity. Illustration of low-rate problem in a sequential-activity network of conductance-based spiking neurons without oscillatory input. Drifting bumps of activity in the network initiated by short (100 ms) pulses (labeled by colored bars) exhibit only a single level of activity. (C) Sequential-activity network with oscillatory input is able to maintain drifting bumps with discretely graded levels of activity.

Methods
The Wang-Buzsaki model neuron used for most spiking neuron simulations in this paper is based upon the original model described in (34). Below, we show the equations for the dynamical variables most relevant to the maintenance of discretely graded persistent activity. The full model equations are included in the Supporting Information. The membrane potential of the Wang-Buzsaki neuron obeys the current balance equation: (2) *+, % = − % + *+, 5 9 − % *4%50 : ( ) = ( ) (4) where ℎ and are time-varying channel variables (Supporting Information). The parameter values used are specified in Tables S1 and S2. The Wang-Buzsaki neuron receives several sources of inputs: (1) *+,,% ( . , … , & ) represents recurrent feedback to neuron , the strength of which is determined by a weight matrix %2 defining the strength of the connection from neuron to neuron , (2) / is a constant current that shifts the resting potential, and could represent tonic background input or intrinsic currents not explicitly modeled, (3) ( ) is the external oscillatory input ( ( ) = 0 for models with no oscillatory input), and (4) 01#,% ( ) represents the external inputs to be accumulated and stored by the memory network. To calculate spike times in equation 3, we used the time of the peak of the action potential, with only action potentials exceeding a voltage of 0 mV included. Integration was performed numerically using the fourth order Runge-Kutta method with a time step ∆t = 10 −2 ms.
In the single neuron case, there is a single recurrent synaptic weight, [ / 6 ]. Values for all simulation parameters are included in Table S2. In Figures 4-6, we study three different network architectures composed of Wang-Buzsaki neurons: an all-to-all connectivity (Figure 4), a ring structure (Figure 5), and a directed structure ( Figure 6).
The all-to-all connected networks of Figure 4 Figure 4D implements a network in which these uniform connection strengths have been perturbed by adding static Gaussian noise of mean zero independently to each connection. Exponentially filtered temporally white noise (Ornstein-Uhlenbeck process) input was implemented in the network illustrated in Figure 4C; for each neuron , the additive noise was given independently by: %,# ∼ (0,1) (6) where , is the standard deviation of the noise.
For the ring connectivity structure in Figure 5, the connection strength from neuron to neuron is described by: The directed structure illustrated in Figure 6 resembles the ring structure, but results in a drift of the 'activity bump' in one direction. The connection strength from neuron j to neuron i in this case is defined by: where is the Heaviside (step) function and C controls the spatial extent of the connectivity.

Comparison to linear spiking autapse model
In Figure 3, we compare the robustness of discretely graded persistent activity of the phase-locking nonlinear spiking model described above, to that of a spiking autapse model in which analog persistent activity is enabled by excitatory feedback that is tuned to offset the intrinsic decay of activity. The autapse model is described in detail in (37); equations describing the dynamics of the model are included in the Supporting Information.

Simple rate model
The equation for the simple rate model implemented for Figure 1 is given in Figure 1A. The nonlinear term used for Figure 1B , ⌧ n (V ) = 1 ↵ n (V ) + n (V )