Abstract
In sensory circuits with poor feature topography, stimulus-specific feedback inhibition necessitates carefully tuned synaptic circuitry. Recent experimental data from mouse primary visual cortex (V1) show that synapses between pyramidal neurons and parvalbumin-expressing (PV) inhibitory interneurons tend to be stronger for neurons that respond to similar stimulus features. The mechanism that underlies the formation of such excitatory-inhibitory (E/I) assemblies is unresolved. Here, we show that activity-dependent synaptic plasticity on input and output synapses of PV interneurons generates a circuit structure that is consistent with mouse V1. Using a computational model, we show that both forms of plasticity must act synergistically to form the observed E/I assemblies. Once established, these assemblies produce a stimulus-specific competition between pyramidal neurons. Our model suggests that activity-dependent plasticity can enable inhibitory circuits to actively shape cortical computations.
Introduction
With the advent of modern optogenetics, the functional role of inhibitory interneurons has developed into one of the central topics of systems neuroscience [Fishell and Kepecs, 2019]. Aside from the classical perspective that inhibition serves to stabilize recurrent excitatory feedback loops in neuronal circuits [van Vreeswijk and Sompolinsky, 1996, Brunel, 2000, Murphy and Miller, 2009, Sprekeler, 2017], it is increasingly recognised as an active player in cortical computation [Isaacson and Scanziani, 2011, Priebe and Ferster, 2008, Rubin et al., 2015, Pouille and Scanziani, 2001, Letzkus et al., 2011, Adesnik et al., 2012, Hennequin et al., 2014, Phillips et al., 2017, Barron et al., 2016, 2017, Tovote et al., 2015].
Within cortical neurons, excitatory and inhibitory currents are often highly correlated in their response to stimuli [Wehr and Zador, 2003, Froemke et al., 2007, Tan et al., 2011, Bhatia et al., 2019], in time [Okun and Lampl, 2008, Dipoppa et al., 2018] and across neurons [Xue et al., 2014]. This cotuning of excitatory and inhibitory currents has been attributed to different origins. In topographically organised sensory areas such as cat primary visual cortex, the co-tuning with respect to sensory stimuli could be a natural consequence of local feedback inhibition and does not impose strong constraints on inhibitory circuitry [Harris and Mrsic-Flogel, 2013]. In the case of feedforward inhibition, co-tuning of excitatory and inhibitory currents was suggested to arise from homeostatic synaptic plasticity in GABAergic synapses [Vogels et al., 2011, Clopath et al., 2016, Weber and Sprekeler, 2018, Hennequin et al., 2017].
In sensory areas with poor feature topography, such as primary visual cortex of rodents [Ohki et al., 2005], feedback inhibition has been hypothesised to be largely unspecific for stimulus features, a property inferred from the dense connectivity [Fino and Yuste, 2011, Packer and Yuste, 2011] and reliable presence of synapses connecting pyramidal (Pyr) neurons to inhibitory interneurons with dissimilar stimulus tuning [Harris and Mrsic-Flogel, 2013, Bock et al., 2011, Hofer et al., 2011]. However, recent results cast doubt on this idea of a “blanket of inhibition” [Fino and Yuste, 2011, Packer and Yuste, 2011].
In mouse primary visual cortex (V1), Znamenskiy et al. [2018] report that although the presence of synaptic connections between Pyr cells and parvalbumin-positive (PV) interneurons is independent of their respective stimulus responses, the efficacy of those synapses is correlated with their response similarity, both in PV → Pyr and in Pyr → PV connections. These mutual preferences in synaptic organization suggest that feedback inhibition may be more stimulus-specific than previously thought and that Pyr and PV neurons form specialized—albeit potentially overlapping—excitatory-inhibitory (E/I) assemblies [Chenkov et al., 2017, Yoshimura et al., 2005, Litwin-Kumar and Doiron, 2012, 2014]. While the presence of such E/I assemblies [Znamenskiy et al., 2018, Rupprecht and Friedrich, 2018] suggests the need for an activity-dependent mechanism for their formation and/or refinement [Khan et al., 2018, Najafi et al., 2020], the requirements such a mechanism must fulfil remain unresolved.
Here, we use a computational model to identify requirements for the development of stimulus-specific feedback inhibition. We find that the formation of E/I assemblies requires a synergistic action of plasticity on two synapse types: the excitatory synapses from Pyr neurons onto PV interneurons, and the inhibitory synapses from those interneurons onto the Pyr cells. Using “knock-out experiments”, in which we block plasticity in either synapse type, we show that both must be plastic to account for the observed functional microcircuits in mouse V1. In addition, after the formation of E/I assemblies, perturbations of individual Pyr neurons lead to a feature-specific suppression of other Pyr neurons as recently found in mouse V1 [Chettih and Harvey, 2019]. Thus, synergistic plasticity of the in- and outgoing synapses of PV interneurons can drive the development of stimulus-specific feedback inhibition, resulting in a competition between Pyr neurons with similar stimulus preference.
Results
To understand which activity-dependent mechanisms can generate specific feedback inhibition in circuits without feature topography—such as mouse V1 (Fig. 1a), we studied a rate-based network model consisting of N E = 512 excitatory Pyr neurons and N I = 64 inhibitory PV neurons. To endow the excitatory neurons with a stimulus-tuning similar to pyramidal cells in layer 2/3 of mouse V1 [Znamenskiy et al., 2018], each excitatory neuron receives external excitatory input that is tuned to orientation, temporal frequency and spatial frequency (Fig. 1b). The preferred stimuli of the Pyr neurons cover the stimulus space evenly. Because we are interested under which conditions feedback inhibition can acquire a stimulus-selectivity, inhibitory neurons receive external inputs without stimulus tuning, but are recurrently connected to Pyr neurons. While the network has no stimulus topography, Pyr neurons are preferentially connected to other Pyr neurons with similar stimulus tuning, and connection strength is proportional to the signal correlation of their external inputs. Connection probability across the network is p = 0.6, with the remaining network connectivity (Pyr → PV, PV → PV, PV → Pyr) initialised randomly according to a log-normal distribution [Song et al., 2005, Loewenstein et al., 2011], such that feedback inhibition is not stimulus specific at the outset.
E/I assemblies are formed by homeostatic plasticity rules in input and output connections of PV interneurons
In feedforward networks, a stimulus-specific balance of excitation and inhibition can arise from homeostatic inhibitory synaptic plasticity that aims to minimise the deviation of a neuron’s firing rate from a target for all stimuli of a given set [Vogels et al., 2011, Clopath et al., 2016, Weber and Sprekeler, 2018]. We wondered whether a stimulus-specific form of homeostasis can also generate stimulus-specific feedback inhibition by forming E/I assemblies. To that end, we derive gradient-based synaptic plasticity rules for excitatory input and inhibitory output connections of PV interneurons that are homeostatic for the excitatory population (see Materials & Methods). The resulting gradient-based rules essentially constitute a backpropagation of error [Rumelhart et al., 1985] through the network. Because back-propagation rules rely on non-local information that might not be available to the respective synapses, their biological plausibility is currently debated [Lillicrap et al., 2020, Sacramento et al., 2018, Guerguiev et al., 2017, Whittington and Bogacz, 2019]. Consequently, we use local approximations of the derived rules (see Supplementary Materials). The resulting learning rule for the output connections of the interneurons is similar to a previously suggested form of homeostatic inhibitory plasticity (Supp. Fig. S1a, left) [Vogels et al., 2011]. Specifically, PV output synapses undergo Hebbian changes in proportion to presynaptic interneuron activity and the signed deviation of postsynaptic pyramidal cell activity from the homeostatic target. In contrast, the PV input synapses are changed such that the total excitatory drive from the Pyr population to each interneuron is close to some target value (Supp. Fig. S1a, right). Such a rule effectively implements a recurrent variant of feedback alignment [Akrout et al., 2019, Lillicrap et al., 2016]. While our main results are obtained using the local approximations, we also simulated the gradient-based rules to verify that the approximation does not qualitatively change the results (Supp. Fig. S4).
When we endow the synapses of an initially randomly connected network of Pyr neurons and PV interneurons with plasticity in both the input and the output synapses of the interneurons, the network develops a synaptic weight structure and stimulus response that closely resemble that of mouse V1 [Znamenskiy et al., 2018]. Before learning, interneurons show poor stimulus selectivity (Fig. 1c), in line with the notion that in a random network, interneurons pool over many Pyr neurons with different stimulus tuning [Harris and Mrsic-Flogel, 2013]. The network is then exposed to randomly interleaved stimuli. By the end of learning, interneurons have developed a pronounced stimulus tuning, albeit weaker than that of Pyr neurons (Fig. 1c, d). Interneurons form strong bidirectional connections preferentially with Pyr neurons with a similar stimulus tuning, whereas connections between Pyr-PV pairs with dissimilar stimulus tuning are weaker (Fig. 1d, e). To make our results comparable to Znamenskiy et al. [2018], we randomly sample an experimentally feasible number of synaptic connections from the network (n = 100). Both the efficacy of PV input and output connections are highly correlated with the response similarity (see Materials & Methods) of the associated Pyr neurons and interneurons (Fig. 1e, left and center). For bidirectionally connected cell pairs, the efficacies of the respective input and output connections are highly correlated (Fig. 1e, right). The stimulus tuning of the inhibitory inputs onto the Pyr cells—initially flat—closely resembles that of the excitatory inputs after learning (Fig. 1f, Supp. Fig. S2) [Tan et al., 2011], i.e. the network develops a precise E/I balance [Hennequin et al., 2017].
Finally, the optimal gradient rules produce very similar results to the local approximations (Supp. Fig. S4). When we compare the weight updates the approximate rules perform to the updates that would occur using the gradient rule, the weight updates of the local approximations align to those of the gradient rules over learning (Fig. 1g, Supp. Fig. S3), presumably by a mechanism akin to feedback alignment [Lillicrap et al., 2016, Akrout et al., 2019].
In summary, these results show that combined homeostatic plasticity in input and output synapses of interneurons can generate a similar synaptic structure as observed in mouse V1, including the formation of E/I assemblies.
PV → Pyr plasticity is required for the formation of E/I assemblies
To better understand the respective roles of the input and output synapses for the formation of E/I assemblies, we perform “knock-out” experiments, in which we selectively block synaptic plasticity in either of the synapse types.
When we knock-out output plasticity but keep input plasticity intact, the network fails to develop E/I assemblies and a stimulus-specific E/I balance. While there is highly significant change in the distribution of PV interneuron stimulus selectivity (Mann–Whitney U test, U = 1207, p < 10−4), the effect is much stronger when output plasticity is also present (Fig. 2a,b). Importantly, excitatory and inhibitory currents in Pyr neurons are poorly co-tuned (Fig. 2c, Supp. Fig. S2b). In particular, feedback inhibition remains largely untuned because output connections are still random, so that Pyr neurons pool inhibition from many interneurons with different stimulus tuning.
To investigate whether the model without output plasticity is consistent with the synaptic structure of mouse V1, we repeatedly sample an experimentally feasible number of synapses (n = 100, Fig. 2d) and plot the distribution of the three pairwise Pearson correlation coefficients between the two classes of synaptic weights and response similarity (Fig. 2e). When both forms of plasticity are present in the network, a highly significant positive correlation (p < 0.01) is detected in all samples for all three correlation types (Fig. 2f). When output plasticity is knocked out, we still find a highly significant positive correlation between input weights and response similarity in 99% of the samples (Fig. 2d-f). In contrast, correlations between input and output synapses are weaker and cannot reliably be detected (2% of samples). Notably, we find a correlation between output weights and response similarity in 0.0% of samples (Fig. 2f). Finally, for an experimentally realistic sample size of n = 100, the probability of a correlation coefficient equal or higher than that observed by Znamenskiy et al. [2018] is 0.0% for the correlation between output weights and response similarity (r = 0.55), and 0.0% for the correlation between input and output synapses (r = 0.52).
The non-local gradient rule for the PV input synapses alone also does not permit the formation of E/I assemblies (Supp. Fig. S4a). While the selectivity of interneurons increases more than for the local approximation (Supp. Fig. S4b), feedback inhibition still remains untuned in the absence of output plasticity (Supp. Fig. S4c,d).
We therefore conclude that input plasticity alone is insufficient to generate the synaptic microstructure observed in mouse V1.
Pyr → PV plasticity is required for assembly formation
When we knock out input plasticity but keep output plasticity intact, we again observe no formation of E/I assemblies. This remains true even when using the gradient-based rule (Supp. Fig. S4). The underlying reason is that input weights remain random. Interneurons collect excitation from many Pyr neurons with different preferences, and absent plasticity on their input synapses, they maintain their initial poor stimulus selectivity (Fig. 3a-c). Because of the poor stimulus tuning of the interneurons, output plasticity cannot generate stimulus-specific inhibitory inputs to the Pyr neurons (Fig. 3d). Across the whole population, the similarity of excitatory and inhibitory currents onto Pyr neurons remains low (Supp. Fig. S2b,c).
Note that interneurons still possess a weak, but consistent stimulus tuning that arises from random variations in their input weights. A particularly strong input connection will cause the postsynaptic interneuron to prefer similar stimuli to the presynaptic Pyr. Because of the resulting correlated activity, the Hebbian nature of the output plasticity potentiates inhibitory weights for such cell pairs that are reciprocally connected. This tendency of strong input synapses to generate a strong corresponding output synapse is reflected in a positive correlation between them (Fig. 3e, Supp. Fig. S5a), despite the fact that input synapses remain random.
Collectively, these results indicate that plasticity of both the inhibitory output and the excitatory input synapses of PV interneurons is required for the formation of E/I assemblies in cortical areas without feature topography, such as mouse V1.
Single Neuron Perturbations
Our findings demonstrate that in networks without feature topography, only a synergy of excitatory and inhibitory plasticity can account for the emergence of E/I assemblies. But how does stimulus-specific feedback inhibition affect interactions between excitatory neurons? In layer 2/3 of V1 similarly tuned excitatory neurons tend to have stronger and more frequent excitatory connections [Ko et al., 2011]. It has been hypothesised that this tuned excitatory connectivity supports reliable stimulus responses by amplifying the activity of similarly tuned neurons [Cossell et al., 2015]. However, the presence of co-tuned feedback inhibition could also induce the opposite effect, such that similarly tuned excitatory neurons are in competition with each other [Chettih and Harvey, 2019, Moreno-Bote and Drugowitsch, 2015].
To investigate the effect of stimulus-specific inhibition in our network, we simulate the perturbation experiment of Chettih and Harvey [2019]: First, we again expose the network to the stimulus set, with PV input and output plasticity in place to learn E/I assemblies. Second, both before and after learning, we probe the network with randomly selected stimuli from the same stimulus set, while perturbing a single Pyr cell with additional excitatory input, and measure the resulting change in activity of other Pyr neurons in the network (Fig. 4a).
While the activity of the perturbed neuron increases, many of the other Pyr neurons are inhibited in response to the perturbation (Fig. 4b). Although comparing the pairwise influence of Pyr neurons on each other does not reveal any apparent trend (Fig. 4c), recent experiments report that the influence a single-cell perturbation has on other neurons depends on the similarity of their stimulus feature tuning [Chettih and Harvey, 2019]. To test whether we observe the same feature-specific suppression, we compute the influence of perturbing a Pyr on the rest of the network as a function of the receptive field correlation of the perturbed cell and each measured cell. In line with recent perturbation studies [Chettih and Harvey, 2019, Sadeh and Clopath, 2020], we observe that—on average—neurons are more strongly inhibited if they have a similar tuning to the perturbed neuron (Fig. 4d). The opposite holds before learning: the effect of single-neuron perturbations on the network is increasingly excitatory as receptive field correlation increases. Notably, the networks in which input or output plasticity was knocked out during learning (and therefore did not develop E/I assemblies) show the same excitatory effect (Fig. 4d, Supp. Fig. S6b). This confirms that a “blanket of inhibition” does not account for feature-specific suppression between excitatory neurons [Sadeh and Clopath, 2020].
To better understand this behaviour, we use the Pyr-Pyr receptive field correlations to compute the coefficient of determination for all pairs (R2, which quantifies how well the receptive field of one Pyr neuron predicts that of another). Learning changes the correlative structure in the network (Supp. Fig. S6a), and thereby decreases the coefficient of determination on average, indicating a reduction in Pyr-Pyr correlations within the network (E[R2] = 0.06 before learning, 0.02 after). Thus, plasticity suppresses some of the strongest correlations, resulting in “feature competition” which is believed to aid sensory processing [Lochmann et al., 2012, Moreno-Bote and Drugowitsch, 2015].
While on average the network exhibits feature competition, the influence of individual Pyr neurons on the rest of the network is highly variable. According to recent modeling work [Sadeh and Clopath, 2020], the strength of Pyr → PV synapses strongly influences whether a network will exhibit feature competition. In our network, the total out-going weight of a Pyr cell onto the PV neurons indeed predicts the average influence that neuron will have on the rest of the network when perturbed (Fig. 4e; r = −0.6).
In summary, the stimulus-specific feedback inhibition that emerges in the model also captures the paradoxical suppression of similarly tuned excitatory neurons observed in single-cell perturbation experiments.
Discussion
The idea that feedback inhibition serves as a “blanket of inhibition” [Packer and Yuste, 2011, Fino and Yuste, 2011] that can be selectively broken [Karnani et al., 2016] has been gradually relaxed over recent years and replaced by the notion that feedback inhibition can be rather selective [Rupprecht and Friedrich, 2018] and could thereby support specific neuronal computations [Vogels and Abbott, 2009, Hennequin et al., 2014, Denève and Machens, 2016, Najafi et al., 2020], even in networks without topographic organisation [Znamenskiy et al., 2018, Rupprecht and Friedrich, 2018]. Here, we used a computational model to show that the development of E/I assemblies similar to those observed in mouse V1 [Znamenskiy et al., 2018] or zebrafish olfactory areas [Rupprecht and Friedrich, 2018] can be driven by a homeostatic form of plasticity of the in- and outgoing synapses of inhibitory interneurons. Based on the results of virtual knock-out experiments we suggest that, on their own, input or output plasticity of interneurons are insufficient to explain the Pyr-PV microcircuitry in mouse V1 and that input and output plasticity in interneurons must act in synergy for stimulus-specific feedback inhibition to develop. To investigate how the presence of E/I assemblies affects interactions between excitatory neurons, we mimicked a perturbation experiment and found that—as in mouse visual cortex—stimulating single excitatory cells paradoxically suppresses similarly tuned neurons [Chettih and Harvey, 2019]. Our findings suggest that, by driving the development of tuned feedback inhibition, plasticity of interneurons can fundamentally shape cortical processing.
The learning rules for the input and output synapses of PV interneurons are based on a single homeostatic objective that aims to keep the net synaptic current onto Pyr neurons close to a given target for all stimuli. The two forms of plasticity fulfil different purposes, however. Plasticity of input synapses is required for interneurons to acquire a stimulus selectivity, whereas plasticity of output synapses can exploit interneuron selectivity to shape inhibitory currents onto excitatory cells. The output plasticity we derived for our recurrent network is very similar to a previously suggested form of inhibitory plasticity [Vogels et al., 2011, Sprekeler, 2017]. Homeostatic plasticity rules for inhibitory synapses are now used regularly in computational studies to stabilise model circuits [Vogels et al., 2011, Hennequin et al., 2017, Landau et al., 2016]. In contrast, a theoretically grounded approach for the plasticity of excitatory input synapses onto inhibitory neurons is missing.
Homeostatic changes in excitatory synapses onto interneurons in response to lesions or sensory deprivation have been reported [Keck et al., 2011, Takesian et al., 2013, Kuhlman et al., 2013], but the specific mechanisms and functions of this form of interneuron plasticity are not resolved. The plasticity rule we derived for the input synapses of interneurons effectively changes the selectivity of those neurons according to the demands of the Pyr cells, i.e. such that the interneurons can best counteract deviations of Pyr activity from the target. By which mechanisms such a (nearly teleological) form of plasticity can be achieved is at its core a problem of credit assignment, whose biological implementation remains open [Lillicrap et al., 2016, Guerguiev et al., 2017, Sacramento et al., 2018].
Here, we used a local approximation of the gradient, backpropagation rules, which produces qualitatively similar results, and which we interpret as a recurrent variant of feedback alignment [Lillicrap et al., 2016, Akrout et al., 2019]. The excitatory input connections onto the interneurons serve as a proxy for the transpose of the output connections. The intuition why this replacement is reasonable is the following: The task of balancing excitation by feedback inhibition favours symmetric connections, because excitatory cells that strongly drive a particular PV interneuron should receive a strong feedback connection in return. Therefore, E/I balance favours a positive correlation between the incoming and outgoing synapses of PV neurons and thus the two weight matrices will be aligned in a final balanced state [Lillicrap et al., 2016, Akrout et al., 2019]. This weight replacement effectively replaces the “true” feedback errors by a deviation of the total excitatory input to the PV neurons from a target [Hertäg and Sprekeler, 2020]. The rule therefore has the structure of a homeostatic rule for the recurrent excitatory drive received by PV neurons.
If excitatory inputs to Pyr neurons are much larger than required to reach the target, the homeostatic objective of bringing net currents to that target effectively requires a balance of excitation and inhibition on a stimulus-by-stimulus basis, with a small overshoot of excitation (or, in spiking networks, membrane potential fluctuations) that allows Pyr neurons to fire at the target rate. We speculate that E/I assemblies could be learned not only from the homeostatic objective used here, but by any other objective that enforces a positive correlation of the stimulus tuning of excitatory and inhibitory inputs to neurons in the circuit.
Given that the role of the input plasticity in the interneurons is the formation of a stimulus specificity, it is tempting to assume that could equally well be achieved by classical forms of plasticity like the Bienenstock-Cooper-Munro (BCM) rule [Bienenstock et al., 1982], which are commonly used in models of receptive field formation. In our hands, this approach did not succeed, for the following reason. BCM rules tend to make the postsynaptic neuron as stimulus-selective as possible. Given the limited number of interneurons in our circuit, this can lead to a situation in which parts of stimulus space are not represented by any interneurons. As a result, Pyr neurons that respond to those stimuli cannot recruit inhibition and maintain a high firing rate far above the target. Other Pyr cells, which have access to interneurons with a similar stimulus tuning, can recruit inhibition to gradually reduce their firing rates towards the target rate. Because the BCM rule is Hebbian, it tends to strengthen input synapses from Pyr neurons with high activity. This shifts the stimulus tuning of the interneurons to those stimuli that were previously underrepresented. However, this in turn renders a different set of stimuli uncovered by inhibition and withdraws feedback inhibition from the corresponding set of Pyr cells, which can now fire at high rates. These networks did not reach a steady state (unpublished observations). We suspect that this instability can also arise for other Hebbian forms of plasticity in interneuron input synapses when they are combined with homeostatic inhibitory plasticity [Vogels et al., 2011] in their output synapses.
Like all computational models, the present one contains simplifying design choices. First, we did not include stimulus-specific feedforward inhibition, because the focus lay on the formation of stimulus-specific feedback inhibition. The model could be enriched by feedforward inhibition in different ways. In particular, we expect that the two forms of plasticity will establish E/I assemblies even in the presence of stimulus-selective external inputs to the interneurons, because stimulus-specific external excitation should always be more supportive of the homeostatic objective than unspecific inputs. It may be worth exploring whether adding feedforward inhibition leaves more room for replacing the PV input plasticity that we used by classical Hebbian rules, because the activity of the external inputs remains unaltered by the plasticity in the network (such that the complex instability described above may be mitigated). Given that the focus of this work was on feedback inhibition, an extensive evaluation of the different variants of feedforward inhibition are beyond the scope of the present article.
Second, we neglected much of the complexity of cortical interneuron circuits by including only one class of interneurons. We interpret these interneurons as PV-expressing interneurons, given that PV interneurons provide local feedback inhibition [Hu et al., 2014] and show a stimulus-selective circuitry akin to E/I assemblies [Znamenskiy et al., 2018]. With their peri-somatic targets on Pyr cells, PV-expressing (basket) cells are also a prime candidate for the classical feedback model of E/I balance [van Vreeswijk and Sompolinsky, 1996]. Note that our results do not hinge on any assumptions that are specific to PV neurons, and may thus also hold for other interneuron classes that provide feedback inhibition [Tremblay et al., 2016]. Given that the division of labour of the various cortical interneuron classes is far from understood, an extension to complex interneuron circuits [Litwin-Kumar et al., 2016, Hertäg and Sprekeler, 2019, 2020] is clearly beyond the present study.
Similarly tuned pyramidal cells tend to be recurrently connected [Cossell et al., 2015, Harris and Mrsic-Flogel, 2013], in line with the notion that excitatory cells with similar tuning mutually excite each other. This notion is questioned by a recent perturbation experiment demonstrating feature-specific suppression between pyramidal cells with similar tuning [Chettih and Harvey, 2019]. It has been suggested that this apparently paradoxical effect requires strong and tuned connections between excitatory and inhibitory neurons [Sadeh and Clopath, 2020]. The E/I assemblies that develop in our model provide sufficiently strong and specific inhibitory feedback to cause a suppression between similarly tuned Pyr neurons in response to perturbations. Hence, despite the presence of stimulus-specific excitatory recurrence, Pyr neurons with similar stimulus preference effectively compete. Computational arguments suggest that this feature competition may be beneficial for stimulus processing, e.g. by generating a sparser and more efficient representation of the stimuli [Olshausen and Field, 2004, Denéve and Machens, 2016].
In addition to predicting that knocking out plasticity of inhibitory input or output synapses should prevent the development of E/I assemblies, our model also predicts different outcomes for single neuron perturbation experiments in juvenile and adult mice. Given that in rodents, stimulus-tuning of inhibitory currents occurs later in development than that of excitation [Dorrn et al., 2010], we expect that in juvenile mice single-cell perturbations would not cause feature-specific suppression but amplification due to excitatory recurrence and unspecific feedback inhibition.
Materials & Methods
Network & stimuli
We use custom software to simulate a rate-based recurrent network model containing N E = 512 excitatory and N I = 64 inhibitory neurons. The activation of the neurons follows Wilson-Cowan dynamics:
Here, rE = [hE]+, rI = [hI]+ denote the firing rates of the excitatory and inhibitory neurons, which are given by their rectified activation. WY←X denotes the matrix of synaptic efficacies from population X to population Y (X, Y ∈ {E, I}). The external inputs I(s) to the excitatory neurons have a bell-shaped tuning in the three-dimensional stimulus space consisting of spatial frequency, temporal frequency and orientation [Znamenskiy et al., 2018]. To avoid edge effects, the stimulus space is periodic in all three dimensions, with stimuli ranging from -π to π. The stimulus tuning of the external inputs is modeled by a von Mises function with a maximum of 50 Hz and a tuning width κ = 1. The preferred stimuli of the N E = 512 excitatory cells cover the stimulus space evenly on a 12 × 12 × 12 grid. All neurons receive a constant background input of Ibg = 5 Hz.
Recurrent connections W E←E among excitatory neurons have synaptic weight between neurons i and j that grows linearly with the signal correlation of their external inputs:
The cropping threshold C is chosen such that the overall connection among the excitatory neurons probability is 0.6. The remaining synaptic connections (E→I, I→E, I→I) are initially random, with a connection probability p = 0.6 and log-normal weights. For parameters please refer to Table 1.
During learning, we repeatedly draw all 12 × 12 × 12 preferred stimuli of the Pyr neurons, in random order. This procedure is repeated 500 times to ensure convergence of synaptic weights. To reduce simulation time, we present each stimulus long enough for all firing rates to reach steady state and only then update the synaptic weights.
Synaptic plasticity
The PV → Pyr and Pyr → PV synapses follow plasticity rules that aim to minimize the deviation of the excitatory activations from a target rate ρ0 (ρ0 = 1 Hz): where ⟨. ⟩s denotes the average over all stimuli. When plastic, synaptic weights change according to
After every update of the Pyr → PV matrix, the incoming weights for each PV interneuron are multiplicatively scaled such that their sum is J I←E [Akrout et al., 2019]. In that case, the rule in Eq. (4b) is approximately local in that it compares the excitatory input current received by the postsynaptic PV neuron to a target value I0 = J I←Eρ0, and adjusts the incoming synapses in proportion to this error and to presynaptic activity [see Eq. (4c)].
Both plasticity rules are approximations of the gradient of the objective function Eq. (3). Interested readers are referred to the supplementary methods for their mathematical derivation. For the results in Supp. Fig. S4, we use the Adaptive Moment Estimation (Adam) algorithm [Kingma and Ba, 2014] to improve optimisation performance.
We used a standard reparameterization method to ensure the sign constraints of an E/I network. Moreover, all weights are subject to a small weight-dependent decay term, which aids to keep the firing rates of the interneurons in a reasonable range. For details, please refer to the Supplementary Methods. The learning rule Eq. (4a) for the output synapses of the inhibitory neurons is similar to the rule proposed by Vogels et al. [2011], wherein each inhibitory synapse increases in strength if the deviation of the postsynaptic excitatory cell from the homeostatic target ρ0 is positive (and decreases it when negative). In contrast, the learning rule Eq. (4b) increases activated input synapses for an interneuron if the weighted sum of deviations in its presynaptic excitatory population is positive (and decreases them if it is negative). Though it is local, when operating in conjunction with the plasticity of Eq. (4a), this leads to feedback alignment in our simulations, and effectively performs backpropagation without the need for weight transport [Akrout et al., 2019].
Note that the objective function Eq. (3) can also be interpreted differently. The activation hE of a neuron is essentially the difference between its excitatory and inhibitory inputs. Therefore, the objective function Eq. (3) is effectively the mean squared error between excitation and inhibition, aside from a small constant offset ρ0. The derived learning rules can therefore be seen as supervised learning of the inhibitory inputs, with excitation as the label. They hence aim to establish the best co-tuning of excitation and inhibition that is possible given the circuitry.
Perturbation experiments
The perturbation experiments in Fig. 4 are performed in a network in which both forms of plasticity have converged. The network is then exposed to different stimuli, while the afferent drive to a single excitatory cell i is transiently increased by ΔI = 10 Hz. For each stimulus, we compute the steady state firing rates rj of all excitatory cells both with and without the perturbation. The influence of the perturbation of neuron i on neuron j is defined as the difference between these two firing rates, normalized by the pertubation magnitude [Sadeh and Clopath, 2020]. This stimulation protocol is repeated for 90 randomly selected excitatory neurons. The dependence of the influence on the tuning similarity (Fig. 4d) is obtained by binning the influence of the perturbed neuron i and the influenced neuron j according to their stimulus response correlation, and then averaging across all influences in the bin. During the perturbation experiments, synaptic plasticity was disabled.
Quantitative measures
The response similarity (RS) of the stimulus tuning of two neurons i and j is measured by the dot product of their steady state firing rates in response to all stimuli, normalized by the product of their norms [Znamenskiy et al., 2018]:
The same measure is used for the similarity of synaptic currents onto excitatory neurons in Supp. Fig. S2c & S4d.
There is no structural plasticity, i.e. synapses are never added or pruned. However, when calculating Pearson’s correlation between synaptic weights and RS, we exclude synapses that are too weak to be detected using the experimental protocol employed by Znamenskiy et al. [2018]. The threshold values θE←I & θI←E were chosen to be approximately four orders of magnitude weaker than the strongest synapses in the network. The rules that we investigate here tend to produce bimodal distributions of weights, with the lower mode well below this threshold (Supp. Fig. S7).
The stimulus selectivity of the neurons is measured by the skewness of their response distribution across all stimuli: where . Both the response similarity Eq. (5) and the stimulus selectivity Eq. (6) are adapted from Znamenskiy et al. [2018].
Finally, the angle θ between the gradient G from Eq. (15) and its approximation A from Eq. (4) is given by:
Supplementary Information is available for this paper.
Correspondence and requests for materials should be addressed to H.S.
Contributions
O.M. & H.S. conceived the model. O.M. wrote the simulator, and performed all of the simulations. H.S. supervised the project, and acquired the funding. All authors contributed to the experimental design, interpretation of results, and writing of the manuscript.
Competing Interests
The authors declare no competing interests.
Supplementary Materials
Plasticity rules
The general framework we follow to derive homeostatic rules is to minimise the mean squared deviation of individual excitatory (Pyr) neuron activations from a target for all stimuli. More specifically, we perform gradient descent on the following objective function:
Note that the activations hE are given by the difference between the excitatory and the inhibitory inputs to the excitatory neurons. Our approach can hence be interpreted as supervised learning of the inhibitory circuitry, with the goal of minimising the mean squared loss between the inhibitory and the excitatory inputs (plus the constant target ρ0). In this sense, the derived gradient rules aim to generate the best possible E/I balance across stimuli that is possible with the circuitry at hand.
For reasons of readability, we will first simply state the derived rules. The details of their derivation can be found in the following section.
The sign constraints in excitatory-inhibitory networks require all synaptic weights to remain positive. To ensure this, we reparameterised all plastic weights of the network by a strictly positive soft-plus function W = s+(V) = α−1 ln (1 + exp αV) and optimised the weight parameter V by gradient descent.
In summary, the derived learning rules for the synaptic weight parameters between excitatory neuron j and inhibitory interneuron i are given by
Please note that we added a small weight decay to both learning rules. The purpose of this decay term is to avoid an ambiguity in the solution. When the firing rates of the interneurons are increased, but their output weights are decreased accordingly, the firing rates of the excitatory population remain unchanged. Pure gradient-based rules can therefore generate extreme values for the synaptic weights, in which the interneurons have biologically unrealistic firing rates. The additional decay terms in the learning rules solve this issue.
Finally, we replaced the derivative (which should be a Heaviside function, because rates are the rectified activations) by the derivative of a soft-plus function with finite sharpness (α = 1). This allows interneurons to recover from a silent state, in which all gradients vanish. Note that this replacement is done only in the learning rules. The firing rates are still the rectified activations. This method is similar to recent surrogate gradient approaches in spiking networks [Neftci et al., 2019].
Derivation of the homeostatic plasticity rules in recurrent networks
The challenging aspect of the derivation of the learning rules lies in the recurrence of the network. The effects of changes in individual synapses can percolate through the network and thereby change the firing rates of all neurons. Moreover, the temporal dynamics of the network would in principle require a backpropagation of the gradient through time. We circumvent this complication by assuming that the external stimuli to the network change slowly compared to the dynamical time scales of the network, and that the network adiabatically follows the fixed point in its dynamics as the stimulus changes. This assumption significantly simplifies the derivation of the gradient.
The goal is to minimise the total deviation of the excitatory activations hE from the homeostatic target value ρ0. To this end, we calculate the gradient of the objective function in Eq. (3) with respect to a given synaptic weight parameter
We therefore need the gradient of the activations hE of excitatory cells with respect to a parameter v. In the steady state, the activations are given by
The gradient of the activations hE is therefore given by the following implicit condition: where we introduced the diagonal matrices for notational convenience, δij being the Kronecker symbol. Derivatives of expressions that do not depend on any of the synaptic weights in question are excluded.
Eq. (11) requires the gradient of the inhibitory activations with respect to the parameter v, which can be calculated by a similar approach
Introducing the effective interaction matrix ℳ:= 𝕀 + W I←IDI among the interneurons (𝕀 being the identity matrix) allows to solve for the gradient of hI:
Inserting this expression into Eq. (11) yields
Introducing the effective interaction matrix 𝒲= 𝕀 − W E←EDE + W E←IDIℳ−1W I←EDE among the excitatory neurons yields an explicit expression for the gradient of hE:
To obtain gradients with respect to a particular network parameter, we simply substitute the chosen parameter into Eq. (12). For the parameters of the input synapses to the interneurons, the gradient reduces to and for the parameters of the output synapses from the interneurons we get
By inserting these expressions into Eq. (9) and dropping the average, we obtain online learning rules for the input and output synapses of the interneurons:
Note that the same approach also yields learning rules for the threshold and the gain of the transfer function of the inhibitory interneurons, if those are parameters of the system. Although we did not use such intrinsic plasticity rules, we include them here for the interested reader. We assumed a threshold linear transfer function of the interneurons: , where gi is the gain of the neuronal transfer function and θi a firing threshold. While the firing threshold can become negative, gain is reparameterised via the strictly positive soft-plus .
The gradient-based learning rule for the firing thresholds θi of the interneurons is given by and the corresponding learning rule for the interneuron gain gi is
Approximating the gradient rules
In the gradient-based rules derived in the previous section, the 𝒲−1 and ℳ−1 terms account for the fact that a change in a given synaptic connections percolates through the network. As a result, the learning rules are highly nonlocal and hard to implement in a biologically plausible way. To resolve this challenge, we begin by noting that which holds if . is a matrix that depends on the synaptic weights in the network. A similar relation holds for ℳ−1. Since those matrices are contained in Eq. (15a), we substitute the equivalent sums into the relevant sub-expression and truncate the geometric series after the 0-th order, as in
The truncation to 0-th order in the last line should yield an acceptable approximation if synapses are sufficiently weak. The effect of higher-order interactions in the network can then be ignored. This approximation can be substituted into Eq. (15a) and yields an equation that resembles a backpropagation rule in a feedforward network (E → I → E) with one hidden layer—the interneurons. The final, local approximation used for the simulations in the main text is then reached by replacing the output synapses of the interneurons by the transpose of their input synapses. While there is no mathematical argument why this replacement is valid, it turns out to be in the simulations, presumably because of a mechanism akin to feedback alignment Lillicrap et al. [2016], see discussion in the main text.
The same logic of using a 0-th order approximation of 𝒲−1 that neglects higher order interactions is employed to recover the inhibitory synaptic plasticity rule of Vogels et al. [2011] from Eq. (15b).
Supplementary Figures
Acknowledgements
We thank Joram Keijser for helpful discussions that inspired parts of this work. He, along with Denis Alevi, Loreen Hertäg and Robert T. Lange also provided careful proof-reading of the manuscript. This project was funded by the German Federal Ministry for Science and Education through a Bernstein Award (BMBF, FKZ 01GQ1201) and by the German Research Foundation (DFG, collaborative research center FOR 2143).
Footnotes
Network model has been modified to use a local plasticity rule on Pyr-to-PV synapses. Main text and results of Fig. 1-4 have been updated to reflect that change. Supplementary text and figures have also been updated as appropriate.