Learning spatiotemporal signals using a recurrent spiking network that discretizes time

TL;DR

This paper introduces a biologically plausible spiking recurrent neural network model that learns to encode and reproduce complex spatiotemporal sequences, demonstrating robustness and relevance to behavior.

Contribution

It presents a novel biologically plausible model that learns spatiotemporal sequences using Hebbian rules in a spiking recurrent network.

Findings

The model can learn behaviorally relevant spatiotemporal dynamics.

Learned sequences are robustly replayed during spontaneous activity.

The approach encodes time and space using biologically realistic neurons.

Abstract

Learning to produce spatiotemporal sequences is a common task that the brain has to solve. The same neural substrate may be used by the brain to produce different sequential behaviours. The way the brain learns and encodes such tasks remains unknown as current computational models do not typically use realistic biologically-plausible learning. Here, we propose a model where a spiking recurrent network of excitatory and inhibitory biophysical neurons drives a read-out layer: the dynamics of the driver recurrent network is trained to encode time which is then mapped through the read-out neurons to encode another dimension, such as space or a phase. Different spatiotemporal patterns can be learned and encoded through the synaptic weights to the read-out neurons that follow common Hebbian learning rules. We demonstrate that the model is able to learn spatiotemporal dynamics on time scales that are behaviourally relevant and we show that the learned sequences are robustly replayed during a regime of spontaneous activity.

Figures

Tables

Table 1. Table 1: Initialization of network

Constant	Value	Description
$N^E$	$2400$	Number of recurrent E neurons
$N^I$	$600$	Number of recurrent I neurons
$p$	$0.2$	Recurrent network connection probability
$w_0^{EE}$	$2.83$ p$F$	Initial E to E synaptic strength
$w^{IE}$	$1.96$ p$F$	E to I synaptic strength
$w_0^{EI}$	$62.87$ p$F$	Initial I to E synaptic strength
$w^{II}$	$20.91$ p$F$	I to I synaptic strength
$w_0^{RE}$	$0$ p$F$	Initial E to R synaptic strength
$w^{RS}$	$200$ p$F$	S to R synaptic strength
$w^{RH}$	$200$ p$F$	H to R synaptic strength
$w^{HR}$	$200$ p$F$	R to H synaptic strength

Table 2. Table 2: Neuronal membrane dynamics parameters

Constant	Value	Description
${\tau }_E$	$20$ m$s$	E membrane potential time constant
${\tau }_I$	$20$ m$s$	I membrane potential time constant
${\tau }_{abs}$	$5$ m$s$	Refractory period of E and I neurons
${\tau }_{absR}$	$1$ m$s$	R neurons refractory period
${\tau }_{absS}$	$1$ m$s$	S neurons refractory period
${\tau }_{absH}$	$1$ m$s$	H neurons refractory period
$E^E$	$0$ m$V$	excitatory reversal potential
$E^I$	$-75$ m$V$	inhibitory reversal potential
$E_L^E$	$-70$ m$V$	excitatory resting potential
$E_L^I$	$-62$ m$V$	inhibitory resting potential
$V_r$	$-60$ m$V$	Reset potential (for all neurons the same)
$C$	$300$ p$F$	Capacitance
${\Delta }_T^E$	$2$ m$V$	Exponential slope
${\tau }_T$	$30$ m$s$	Adaptive threshold time constant
$V_T$	$-52$ m$V$	Membrane potential threshold
$A_T$	$10$ m$V$	Adaptive threshold increase constant
${\tau }_a$	$100$ m$s$	Adaptation current time constant
$\beta$	$1000$ p$A$	Adaptation current increase constant of recurrent network neurons

Table 3. Table 3: Synaptic dynamics parameters

Constant	Value	Description
${\tau }_d^E$	$6$ m$s$	E decay time constant
${\tau }_r^E$	$1$ m$s$	E rise time constant
${\tau }_d^I$	$2$ m$s$	I rise time constant
${\tau }_r^I$	$0.5$ m$s$	I rise time constant
$W_{ext}^E$	$1.6$ p$F$	External input synaptic strength to E neurons
$r_{ext}^E$	$4.5$ k$Hz$	Rate of external input to E neurons
$W_{ext}^I$	$1.52$ p$F$	External input synaptic strength to I neurons
$r_{ext}^I$	$2.25$ k$Hz$	Rate of external input to I neurons
$W_{ext}^S$	$1.6$ p$F$	External input synaptic strength to S neurons
$W_{ext}^H$	$1.6$ p$F$	External input synaptic strength to H neurons
$r_{ext}^H$	$1.0$ k$Hz$	Rate of external input to H neurons

Table 4. Table 4: Excitatory plasticity parameters

Constant	Value	Description
$A_{LTD}$	$0.0014$ p$A$ m$V^{-2}$	LTD amplitude
$A$	$0.0008$ p$A$ m$V^{-1}$	LTP amplitude (in RNN: $A_{LTP}$ = $A$)
${\theta }_{LTD}$	$-70$ m$V$	LTD threshold
${\theta }_{LTP}$	$-49$ m$V$	LTP threshold
${\tau }_u$	$10$ m$s$	Time constant of low pass filtered
		postsynaptic membrane potential (LTD)
${\tau }_v$	$7$ m$s$	Time constant of low pass filtered
		postsynaptic membrane potential (LTP)
${\tau }_{xEE}$	$3.5$ m$s$	Time constant of low pass filtered
		presynaptic spike train in recurrent network
${\tau }_{xRE}$	$5$ m$s$	Time constant of low pass filtered
		presynaptic spike train for read-out synapses
$W_{min}^{EE}$	$1.45$ p$F$	Minimum E to E weight
$W_{max}^{EE}$	$32.68$ p$F$	Maximum E to E weight
$W_{min}^{RE}$	$0$ p$F$	Minimum E to R weight
$W_{max}^{RE}$	$25$ p$F$	Maximum E to R weight

Table 5. Table 5: Inhibitory plasticity parameters

Constant	Value	Description
$A_{inh}$	${10}^{-5}$ $AHz$	Amplitude of inhibitory plasticity
$r_0$	$3$ $Hz$	Target firing rate
${\tau }_y$	$20$ m$s$	Time constant of low pass filtered spike train
$W_{min}^{EI}$	$48.7$ p$F$	Minimum I to E weight
$W_{max}^{EI}$	$243$ p$F$	Maximum I to E weight

Table 6. Table 6: Parameters for the large recurrent network (all the other parameters are the same as the smaller network)

Constant	Value	Description
$N^E$	$6400$	Number of recurrent E neurons
$N^I$	$1600$	Number of recurrent I neurons
$w_0^{EE}$	$1.73$ p$F$	baseline E to E synaptic strength
$w^{IE}$	$1.20$ p$F$	E to I synaptic strength
$w_0^{EI}$	$40$ p$F$	Initial I to E synaptic strength
$w^{II}$	$12.80$ p$F$	I to I synaptic strength
$W_{min}^{EE}$	$1.27$ p$F$	Minimum E to E weight
$W_{max}^{EE}$	$30.5$ p$F$	Maximum E to E weight
$W_{min}^{EI}$	$40$ p$F$	Minimum I to E weight
$W_{max}^{EI}$	$200$ p$F$	Maximum I to E weight
$W_{max}^{RE}$	$15$ p$F$	Maximum E to R weight

Equations

$$\frac{d{V}^E}{dt}=\frac{1}{{\tau }^E}\left(E_L^E-V^E+{\Delta }_T^E\exp \left(\frac{V^E-V_T^E}{{\Delta }_T^E}\right)\right)+g^{EE}\frac{E^E-V^E}{C}+g^{EI}\frac{E^I-V^E}{C}-\frac{a^E}{C}$$

$${\tau }_T\frac{d{V}_T^E}{dt}=V_T-V_T^E.$$

$${\tau }_a\frac{d{a}^E}{dt}=-a^E.$$

$$\frac{d{V}^R}{dt}=\frac{1}{{\tau }^E}\left(E_L^E-V^R+{\Delta }_T^E\exp \left(\frac{V^R-V_T^R}{{\Delta }_T^E}\right)\right)+g^{RE}\frac{E^E-V^R}{C}+g^{RS}\frac{E^E-V^R}{C}+g^{RH}\frac{E^I-V^R}{C}$$

$$\frac{d{V}^S}{dt}=\frac{1}{{\tau }^E}\left(E_L^E-V^S+{\Delta }_T^E\exp \left(\frac{V^S-V_T^S}{{\Delta }_T^E}\right)\right)+g^{SE}\frac{E^E-V^S}{C}$$

$$\frac{d{V}^I}{dt}=\frac{E_L^I-V^I}{{\tau }^I}+g^{IE}\frac{E^E-V^I}{C}+g^{II}\frac{E^I-V^I}{C}.$$

$$\frac{d{V}^H}{dt}=\frac{E_L^I-V^H}{{\tau }^I}+g^{HE}\frac{E^E-V^H}{C}$$

$$g_i^{XY}(t)=K^Y(t)\ast \left(W_{ext}^X{s}_{i,ext}^X+\sum _j{W}_{ij}^{XY}{s}_j^Y(t)\right).$$

$$K^Y(t)=\frac{e^{-t/{\tau }_d^Y}-e^{-t/{\tau }_r^Y}}{{\tau }_d^Y-{\tau }_r^Y},$$

$$\frac{dW_{ij}}{dt}=-A_{LTD}\,s_{j}(t)\,R\bigl{(}u_{i}(t)-\theta_{LTD}\bigr{)}+A_{LTP}\,x_{j}(t)\,R\bigl{(}V_{i}(t)-\theta_{LTP}\bigr{)}\,R\bigl{(}v_{i}(t)-\theta_{LTD}\bigr{)}.$$

$${\tau }_u\frac{d{u}_i}{dt}$$

$${\tau }_v\frac{d{v}_i}{dt}$$

$${\tau }_x\frac{d{x}_j}{dt}=s_j-x_j.$$

$$A_{LTP}=A\frac{W_{max}^{RE}-W^{RE}}{W_{max}^{RE}-W_{min}^{RE}}.$$

$$\frac{d{W}_{ij}}{dt}=A_{inh}\left(y_i^E(t)-2r_0{\tau }_y\right)s_j^I(t)+A_{inh}{y}_j^I(t)s_i^E(t)$$

$$W=\left(\begin{array}{cc}{W}^{EE}& W^{IE}\\ -W^{EI}& -W^{II}\end{array}\right).$$

$$\frac{dx}{dt}=-x+Ax+\xi$$

$$A=\left[\begin{array}{cccc}\delta & 1& ϵ& -kw\\ ϵ& \delta & 1& -kw\\ 1& ϵ& \delta & -kw\\ \frac{w}{3}& \frac{w}{3}& \frac{w}{3}& -kw\end{array}\right]$$

$$T=\left[\begin{array}{cccc}-w(k-1)& \sqrt{3}w\left(k+\frac{1}{3}\right)& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \delta -\frac{ϵ+1}{2}& -\frac{\sqrt{3}}{2}(ϵ-1)\\ 0& 0& \frac{\sqrt{3}}{2}(ϵ-1)& \delta -\frac{ϵ+1}{2}\end{array}\right]$$

$${\mathbf{u}}_1=\frac{1}{2}\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right],{\mathbf{u}}_2=\frac{1}{2\sqrt{3}}\left[\begin{array}{c}1\\ 1\\ 1\\ -3\end{array}\right],{\mathbf{u}}_3=\frac{1}{\sqrt{2}}\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\end{array}\right],{\mathbf{u}}_4=\frac{1}{\sqrt{6}}\left[\begin{array}{c}-1\\ 2\\ -1\\ 0\end{array}\right].$$

Full text

Learning spatiotemporal signals using a recurrent spiking network that discretizes time

Amadeus Maes

Department of Bioengineering, Imperial College London, United Kingdom

Mauricio Barahona

Department of Mathematics, Imperial College London, United Kingdom

Claudia Clopath

Department of Bioengineering, Imperial College London, United Kingdom

Abstract

Learning to produce spatiotemporal sequences is a common task that the brain has to solve. The same neural substrate may be used by the brain to produce different sequential behaviours. The way the brain learns and encodes such tasks remains unknown as current computational models do not typically use realistic biologically-plausible learning. Here, we propose a model where a spiking recurrent network of excitatory and inhibitory biophysical neurons drives a read-out layer: the dynamics of the driver recurrent network is trained to encode time which is then mapped through the read-out neurons to encode another dimension, such as space or a phase. Different spatiotemporal patterns can be learned and encoded through the synaptic weights to the read-out neurons that follow common Hebbian learning rules. We demonstrate that the model is able to learn spatiotemporal dynamics on time scales that are behaviourally relevant and we show that the learned sequences are robustly replayed during a regime of spontaneous activity.

Author summary

The brain has the ability to learn flexible behaviours on a wide range of time scales. Previous studies have successfully built spiking network models that learn a variety of computational tasks, yet often the learning involved is not biologically plausible. Here, we investigate a model that uses biological-plausible neurons and learning rules to learn a specific computational task: the learning of spatiotemporal sequences (i.e., the temporal evolution of an observable such as space, frequency or channel index). The model architecture facilitates the learning by separating the temporal information from the other dimension. The time component is encoded into a recurrent network that exhibits sequential dynamics on a behavioural time scale, and this network is then used as an engine to drive the read-out neurons that encode the spatial information (i.e., the second dimension). We demonstrate that the model can learn complex spatiotemporal spiking dynamics, such as the song of a bird, and replay the song robustly spontaneously.

Introduction

Neuronal networks perform flexible computations on a wide range of time scales. While individual neurons operate on the millisecond time scale, behaviour time scales typically span from a few milliseconds to hundreds of milliseconds and longer. Building functional models that bridge this time gap is of increasing interest [Abbott et al., 2016], especially now that the activity of many neurons can be recorded simultaneously [Jun et al., 2017, Maass, 2016]. Many tasks and behaviours in neuroscience consist of learning and producing flexible spatiotemporal sequences, e.g. a 2-dimensional pattern with time on the x-axis and any other observable on the y-axis which we denote here in general terms as the "spatial information". For example, songbirds produce their songs through a specialized circuit: neurons in the HVC nucleus burst sparsely at very precise times to drive the robust nucleus of the arcopallium which in its turn drives motor neurons [Hahnloser et al., 2002, Leonardo and Fee, 2005]. For different motor tasks, sequential neuronal activity is recorded in various brain regions [Pastalkova et al., 2008, Itskov et al., 2011, Harvey et al., 2012, Peters et al., 2014, Adler et al., 2019], and while the different tasks involve different sets of muscles, the underlying computation on a more fundamental level might be similar [Rhodes et al., 2004].

Theoretical and computational studies have shown that synaptic weights of recurrent networks can be set appropriately so that dynamics on a wide range of time scales is produced [Litwin-Kumar and Doiron, 2014, Zenke et al., 2015, Tully et al., 2016]. In general, these synaptic weights are engineered to generate a range of interesting dynamics. In slow-switching dynamics, for instance, the wide range of time scales is produced by having stochastic transitions between clusters of neurons [Schaub et al., 2015]. Another example is sequential dynamics, where longer time scales are obtained by clusters of neurons that activate each other in a sequence. This sequential dynamics can emerge by a specific connectivity in the excitatory neurons [Chenkov et al., 2017, Setareh et al., 2018] or in the inhibitory neurons [Billeh and Schaub, 2018]. However, it is unclear how the brain learns these dynamics, as most of the current approaches use non biologically plausible ways to set or "train" the synaptic weights. For example, FORCE training [Sussillo and Abbott, 2009, Laje and Buonomano, 2013, Nicola and Clopath, 2017] or backpropagation through time [Werbos, 1990] use non-local information either in space or in time to update weights. Such information is not available to the synaptic connection, which only has access to the presynaptic and postsynaptic variables at the current time.

Here, we propose to learn a spatiotemporal task over biologically relevant time scales using a spiking recurrent network driving a read-out layer where the neurons and synaptic plasticity rules are biologically plausible. Specifically, all synapses are plastic under typical spike-timing dependent Hebbian learning rules [Clopath et al., 2010, Litwin-Kumar and Doiron, 2014]. Our model architecture decomposes the problem into two parts. First, we train a recurrent network to generate a sequential activity which serves as a temporal backbone so that it operates as a ’neuronal clock’ driving the downstream learning. The sequential activity is generated by clusters of neurons activated one after the other: as clusters are highly recurrently connected, each cluster undergoes reverberating activity that lasts longer than neural time scale so that the sequential cluster activation is long enough to be behaviourally relevant. This construction allows us to bridge the neural and the behavioural time scales. Second, we use Hebbian learning to encode the target spatiotemporal dynamics in the read-out neurons. In this way, the recurrent network encodes time and the read-out neurons encode ’space’. As discussed above, we use the term ’space’ to denote a temporally-dependent observable, be it spatial position, or phase, or a time-dependent frequency, or a more abstract state-space. Similar to the liquid state-machine, where the activity in a recurrent network is linearly read-out by a set of neurons, we can learn different dynamics in parallel in different read-out populations [Jaeger et al., 2007]. We also show that learning in the recurrent network is stable during spontaneous activity and that the model is robust to synaptic failure.

Results

Model architecture

The model consists of two separate modules: a recurrent network and a read-out layer (Fig 1A). Learning happens in two stages. In the first stage, we learn the weights of the recurrent network so that the network exhibits a sequential dynamics. The ensuing recurrent neuronal network (RNN) effectively serves as a temporal backbone driving the learning of the downstream read-out layer. In the second stage, a target sequence is learned in the read-out layer.

Architecture:

The recurrent network is organized in $C$ clusters of excitatory neurons and a central cluster of inhibitory neurons. All excitatory neurons follow adaptive exponential integrate-and-fire dynamics [Brette and Gerstner, 2005] while all inhibitory neurons follow a leaky integrate-and-fire dynamics. The inhibitory neurons in the RNN prevent pathological dynamics. The aim of this module is to discretize time into $C$ sequential intervals, associated with each of the $C$ clusters. This is achieved by learning the weights of the recurrent network. The neurons in the excitatory clusters then drive read-out neurons through all-to-all feedforward connections. The read-out neurons are not interconnected. The target sequence is learned via the weights between the driver RNN and the read-out neurons.

Plasticity:

In previous models, the learning schemes are typically not biologically plausible because the plasticity depends on non-local information. Here, however, we use the voltage-based STDP plasticity rule in all the connections between excitatory neurons (Fig 1B). This is paired with weight normalization in the recurrent network (Fig 1C) and weight dependent potentiation in the read-out synapses (Fig 1D). Inhibitory plasticity [Vogels et al., 2011] finds good parameters aiding the sequential dynamics (Fig S5).

Learning scheme:

During the first stage of learning, all neurons in each cluster receive the same input in a sequential manner. As a result of this learning stage, the recurrent spiking network displays a sequential dynamics of the $C$ clusters of excitatory neurons. Neurons within each cluster spike over a time interval (while all neurons from other clusters are silent), with the activity switching clusters at points $t=[t_{0},t_{1},...,t_{C}]$ so that cluster $i$ is active during time interval $[t_{i-1},t_{i}]$. Thus, time is effectively discretized in the RNN.

During the second stage of learning, the read-out neurons receive input from a set of excitatory supervisor neurons. The discretization of time enables Hebbian plasticity to form strong connections from the neurons in the relevant time bin to the read-out neurons. For instance, if we want to learn a signal which is ‘on’ during $[t_{i-1},t_{i}]$ and ‘off’ otherwise, a supervisor neuron can activate the read-out neuron during that time interval so that connections from cluster $i$ to the read-out neuron are potentiated through activity (who fires together, wires together). This means that, after learning, the read-out neuron will be activated when cluster $i$ is activated. In general, the read-out layer learns a multivariate signal of time, i.e., the neurons in the read-out layer encode the $D$ different dimensions of the target signal: $t\rightarrow\phi(t)=[\phi_{1}(t),\phi_{2}(t),...,\phi_{D}(t)]$.

A recurrent network that encodes discrete time

We give here further details of the first learning stage, where a recurrent network is trained to produce a sequential dynamics. To this end, we initialize the weight matrix so that each synaptic weight between two neurons is non-zero with probability $p$. The weights that are zero remain zero at all times, i.e. the topology is fixed. We set the initial values of the non-zero weights in the recurrent network such that the dynamics is irregular and asynchronous (i.e., a balanced network, see Methods for details).

We stimulate the $C$ clusters with an external input in a sequential manner (Fig 2A): neurons in cluster $i$ each receive external Poisson spike trains (rate of $18$ k$Hz$ for $10$ m$s$, assuming a large input population). After this, there is a time gap where no clusters receive input ($5$ m$s$). This is followed by a stimulation of cluster $i+1$. This continues until the last cluster is reached and then it links back to the first cluster (i.e. a circular boundary condition). During the stimulation, neurons in the same cluster fire spikes together strengthening the intra-cluster connections bidirectionally through the voltage-based STDP rule [Clopath et al., 2010, Ko et al., 2013]. Additionally, there is a pre/post pairing between adjacent clusters. Neurons in cluster $i+1$ fire after neurons in cluster $i$. The weights from cluster $i$ to cluster $i+1$ strengthen unidirectionally (Fig 2B). If the time gap between sequential stimulations is increased during the training phase, so that the gap becomes too long with respect to the STDP time window, then there is no pre/post pairing between clusters and the ensuing dynamics loses its sequential nature and becomes a slow-switching dynamics [Litwin-Kumar and Doiron, 2014, Schaub et al., 2015] (Fig S1). In slow-switching dynamics, clusters of neurons are active over long time scales, but both the length of activation and the switching between clusters is random. This is because the outgoing connections from cluster $i$ to all other clusters are the same in a slow-switching network. To summarize, a connectivity structure emerges through biophysically plausible potentiation by sequentially stimulating the clusters of excitatory neurons in the recurrent network. When the gap between activation intervals is sufficiently small compared to the STDP window, the connectivity structure is such that intra-cluster weights and the weights from successive clusters $i\to i+1$ are strong.

After the synaptic weights have converged, the external sequential input is shut-down and spontaneous dynamics is simulated so that external excitatory Poisson spike trains without spatial or temporal structure drive the RNN. Under such random drive, the sequence of clusters reactivates spontaneously and ensures that both the intra-cluster and the connections from cluster $i$ to cluster $i+1$ remain strong. In general, the interaction between plasticity, connectivity and spontaneous dynamics can degrade the learned connectivity and lead to unstable dynamics [Morrison et al., 2007]. To test the stability of the learned connectivity, we track the changes in the off-diagonal weights (i.e. the connections from cluster $i$ to cluster $i+1$). After the external sequential input is shut-down, we copy the weight matrix and freeze the weights of this copy. We run the dynamics of the recurrent network using the copied frozen weights and apply plastic changes to the original weight matrix. This means that we effectively decouple the plasticity from the dynamics. Indeed, when the dynamics is sequential, the off-diagonal structure is reinforced. When the off-diagonal structure is removed from the frozen copied weight matrix, the dynamics is not sequential anymore. In this case, the off-diagonal structure degrades. We conclude that the connectivity pattern is therefore stable under spontaneous dynamics (Fig S2).

We next studied how the spectrum of the recurrent weight matrix is linked to the sequential dynamics. In linear systems, the eigenvalues of the connectivity matrix determine the dynamics of the system. In a nonlinear spiking model, the relationship between connectivity and dynamics is less clear. The connectivity after learning can be seen as a low-dimensional perturbation of a random matrix. Such low-dimensional perturbations create outliers in the spectrum [Tao, 2013] and change the dynamics [Mastrogiuseppe and Ostojic, 2018]. Here, we have carried out a similar spectral analysis to that presented in  [Schaub et al., 2015] (see Figs 7 in the Methods and S2 in the Supplementary material). The weight matrix has most of its eigenvalues in a circle in the complex plane (Fig 2B) with eigenvalues associated both with the balanced nature of the network, but, importantly, also with the sequential structure (Fig 2B). As the temporal backbone develops through learning (as seen in Fig S2), it establishes a spectral structure in which the pairs of leading eigenvalues with large real parts have almost constant imaginary parts.

A simplified analysis of a reduced weight matrix (where nodes are associated with groups of neurons) shows that the imaginary parts of the dominant eigenvalues depend linearly on the strength of the weights from cluster $i$ to cluster $i+1$ (see Methods, Fig 7). Hence for this simplified linearised rate model, this results in an oscillatory dynamics where the imaginary part determines the frequency by which the pattern of activation returns due to the periodic excitation pattern. As shown in [Schaub et al., 2015], these properties of the linear system carry over to the nonlinear spiking model, i.e., the imaginary parts of the eigenvalues with large real parts determine the time scales of the sequential activity (Fig S2).

Under spontaneous activity, each cluster is active for about $15$ ms, due to the recurrent connectivity within the cluster. A large adaptation current counteracts the recurrent reverberating activity to turn off the activity reliably. Therefore, as each cluster is spontaneously active in a sequence, the sequence length reaches behavioural time scales (Fig 2C). In summary, the network exhibits sequential dynamics, serving as a temporal backbone where time is discretized over behavioural time scales.

Learning a non-Markovian sequence

After the sequential temporal backbone is learnt via the RNN, we can then learn a spatiotemporal sequence via the read-out neurons. To achieve this, during the second stage of training, the read-out neurons receive additional input from supervisor neurons and from interneurons (Fig 3A). The supervisor neurons receive an external Poisson input with rate modulated by the target sequence to be learned (Fig 3B).

As a first example, consider a target sequence composed of states $A$, $B$, $C$ activated in the following deterministic order: $ABCBA$. This is a non-Markovian state sequence because the transition from state $B$ to the next state ($A$ or $C$) requires knowledge about the previous state [Brea et al., 2013], a non trivial task that requires information to be stored about previous network states, potentially over long time periods. Previous studies have proposed various solutions for this task [Maass and Markram, 2002, Brea et al., 2013]. However, separating the problem of sequence learning in two stages solves this in a natural way.

The recurrent network trained in the first stage (Fig 2) is used to encode time. The underlying assumption is that a starting signal activates both the first cluster of the recurrent network and the external input to the supervisor neurons, which activate the read-out neurons.

After the training period, the interneurons and supervisor neurons stop firing (Fig 3C) and the target sequence is stored in the read-out weight matrix (Fig 3D). During spontaneous activity, clusters in the RNN reactivate in a sequential manner driving the learned sequence in the read-out neurons. Hence the spike sequence of the read-out neurons is a noisy version of the target signal (Fig 3E). Learning the same target signal several times results in slightly different read-out spike sequences each time (Fig S3). The firing rates of neurons in the read-out corresponds to the target sequence (Fig 3F). In summary, our results show that the model is able to learn simple but non-trivial spatiotemporal signals that are non-Markovian.

Learning sequences in parallel

We next wondered how multiple spatiotemporal signals can be learned. We hypothesized that, once the temporal backbone is established, multiple spatiotemporal sequences can easily be learned in parallel. As an example, we learn two sequences: $ABCBA$ and $DEDED$. Here, $D$ and $E$ denote two additional read-out neurons (Fig 4A). We assume that the model observes each sequence alternately for $2$ seconds at a time (Fig 4B), although in principle it could also been shown simultaneously. After learning, the target sequences are encoded in the read-out weight matrix (Fig 4C). In a regime of spontaneous dynamics the learned sequences can be replayed (Fig 4D). We conclude that multiple sequences can be learned in parallel. Each separate sequence requires a separate set of read-out neurons. As such, the number of read-out neurons required increases linearly with the number of target sequences.

Properties of the model: scaling, robustness and temporal variability

We investigate several scaling properties of the network. We first assess how the sequential dynamics in the RNN depends on the cluster size by increasing the number of excitatory neurons in each cluster ($N_{C}$), preserving the ratio of excitatory to inhibitory neurons ($N_{E}/N_{I}$). To preserve the magnitude of the currents in the network, the sparseness of the connectivity ($p$) also varies with $N_{C}$ such that $pN_{C}$ is constant. The same training protocols are used for each network configuration as described in the Methods. In Fig 5A, we show that for a fixed number of clusters $C$, the mean period of the sequential dynamics exhibited by the RNN is largely independent of cluster size $N_{C}$. If we fix the number of neurons in the RNN, and in this way change the number of clusters $C$, the mean period of the sequential dynamics decreases with increasing cluster size $N_{C}$. We conclude that the sequential dynamics is preserved over a wide range of network configurations. The time scales in the dynamics depend on the number of clusters and network size.

Another way to modulate the period of the sequential dynamics is to change the unstructured Poisson input to the RNN during spontaneous dynamics (i.e., after the first stage of learning). When the rate of the external excitatory input is increased/decreased, the mean period of the sequential dynamics in the RNN decreases/increases (Fig 5B). These results suggest that the network could learn even if the supervisor signal changes in length at each presentation, assuming that both the supervisor and external Poisson input are modulated by the same mechanism.

We next looked at the robustness of the learning of our model under random perturbations and network size. In this context, we consider the effect of cluster size and the deletion of synapses in the read-out layer after learning. We learn the simple $ABCBA$ sequence (Fig 3) in the read-out neurons using a RNN with a fixed number of clusters $C$ but varying the cluster size $N_{C}$. The total learning time ($\Delta_{t}$) is varied with the cluster size, $N_{C}\Delta_{t}$, because smaller clusters learn slower, since smaller clusters need larger read-out synaptic strengths to drive the same read-out neuron. We also eliminate an increasing number of synapses in the read-out layer. Performance is quantified as the number of spikes elicited by the read-out neurons after deletion of read-out synapses normalized by the number of spikes elicited before deletion. Networks with larger clusters show a more robust performance under noise (Fig 5C and Fig S4). These results show that, not surprisingly, larger clusters drive read-out neurons more robustly and learn faster.

We then tested the limits of time discretization in our model. To that end, we hardcoded a recurrent network with clusters as small as one neuron. In that extreme case, our network becomes a synfire chain with a single neuron in every layer [Abeles, 1991]. In this case, randomly removing a synapse in the network will break the sequential dynamics (Fig 5D). Hence, although a spatiotemporal signal can be learned in the read-out neurons, the signal is not stable under a perturbation of the synfire chain. In summary, the choice of cluster size is a trade-off between network size on the one hand and robustness on the other hand. Large clusters: (i) require a large network to produce sequential dynamics with the same period; (ii) are less prone to a failure of the sequential dynamics; (iii) can learn a spatiotemporal signal faster; and (iv) drive the read-out neurons more robustly.

We have also characterized the variability in the duration of the sequential activity, i.e., the period of the RNN. Since the neural activity does not move through the successive clusters with the same speed in each reactivation, we wondered how the variance in the period of our RNN network compared to Weber’s law. Weber’s law predicts that the standard deviation of reactions in a timing task grows linearly with time [Gibbon, 1977, Hardy and Buonomano, 2018]. Since our model operates on a time scale that is behaviourally relevant, it is interesting to look at how the variability increases with increasing time. Because time in our RNN is discretized by clusters of neurons that activate each other sequentially, the variability increases over time as in a standard Markov chain diffusive process. Hence the variability of the duration $T$ is expected to grow as $\sqrt{T}$ rather than linearly. This is indeed what our network displays (Fig 5E). Here, we scaled the network up and increased the period of the recurrent network by increasing the network size (80 excitatory clusters of 80 neurons each, see Methods for details). By doing so, we can look at how the standard deviation of the duration of the RNN activity grows with time.

Learning a complex sequence

In the non-Markovian target sequence $ABCBA$, the states have the same duration and the same amplitude (Fig 3B). To test whether we could learn more complex sequences, the model was trained using a spatiotemporal signal with components of varying durations and amplitudes. As an example, we use a ’spatio’-temporal signal consisting of a $600$ m$s$ meadowlark song (Fig 6A). The spectrogram of the sound is normalized and used as the time-varying and amplitude-varying rate of the external Poisson input to the supervisor neurons. Each read-out and supervisor neuron encodes a different frequency range, hence in this example our ’space’ dimension is frequency.

We first trained a RNN of $6400$ excitatory neurons (Fig 5E, see Methods) in order to discretize the time interval spanning the full duration of the song. We then trained the read-out layer. The learned read-out weight matrix reflects the structure of the target sequence (Fig 6B). Under spontaneous activity, the supervisor neurons and interneurons stop firing and the recurrent network drives song replays (Fig 6C), and the learned spatiotemporal signal broadly follows the target sequence (Fig 6A). The model performs worse when the target dynamics has time-variations that are faster than or of the same order as the time discretization in the RNN. Thus, we conclude that the model can learn interesting spiking dynamics up to a resolution of time features limited by the time discretization in the recurrent network.

Discussion

We have proposed here a neuronal network architecture based on biophysically plausible neurons and plasticity rules in order to learn spatiotemporal signals. The architecture is formed by two modules. The first module provides a temporal backbone that discretizes time, implemented by a recurrent network where excitatory neurons are trained into clusters that become sequentially active due to strong inter-cluster and cluster $i$ to cluster $i+1$ weights. All of the excitatory clusters are connected to a central cluster of inhibitory neurons. As previously shown for randomly switching clustered dynamics [Litwin-Kumar and Doiron, 2014], the ongoing spontaneous activity does not degrade the connectivity patterns: the set of plasticity rules and sequential dynamics reinforce each other. This stable sequential dynamics provides a downstream linear decoder with the possibility to read out time at behavioural time scales. The second module is a set of read-out neurons that encode another dimension of a signal, which we generically denote as ‘space’ but can correspond to any time-varying observable, e.g., spatial coordinates, frequency, discrete states, etc. The read-out neurons learn spike sequences in a supervised manner, and the supervisor sequence is encoded into the read-out weight matrix. Bringing together elements from different studies [Brea et al., 2013, Nicola and Clopath, 2017, Gilra and Gerstner, 2017], our model exploits a clock-like dynamics encoded in the RNN to learn a mapping to read-out neurons so as to perform the computational task of learning and replaying spatiotemporal sequences. We illustrated the application of our scheme on a simple non-Markovian state transition sequence, a combination of two such simple sequences, and a time series from bird singing with more complex dynamics.

Other studies have focused on the classification of spatiotemporal signals. The tempotron classifies a spatiotemporal pattern by either producing a spike or not [Gütig and Sompolinsky, 2006]. More recent studies on sequential working memory propose similar model architectures that enable the use of Hebbian plasticity [Manohar et al., 2019]. For example, spatiotemporal input patterns can be encoded in a set of feedforward synapses using STDP-type rules [Park et al., 2017, Lee et al., 2019]. Combining these approaches with our model might be an interesting line of future research.

The dynamics of the recurrent network spans three time scales: (i) individual neurons fire at the millisecond time scale; (ii) clusters of neurons fire at the "tick of the clock", $\tau_{c}$, i.e., the time scale that determines the time discretization of our temporal backbone; and (iii) the slowest time scale is at the level of the entire network, i.e. the period of the sequential activity, $\tau_{p}$, achieved over the cascade of sequential activations of the clusters (see Fig 5A). The time scales $\tau_{c}$ and $\tau_{p}$ are dependent on several model parameters: the cluster and network size, the average connection strengths within the clusters, and adaptation. Smaller cluster sizes lead to a smaller $\tau_{c}$ when the network size is fixed and conversely $\tau_{p}$ increases with network size when the cluster size is fixed.

The recurrent network is the "engine" that, once established, drives read-out dynamics. Our model can learn different read-out synapses in parallel (Fig 4) and is robust to synapse failure (Fig 5C). This robustness is a consequence of the clustered organization of the recurrent network. Previously proposed models are also robust to similar levels of noise [Nicola and Clopath, 2017, Nicola and Clopath, 2019]. While an exact comparison is hard to draw, we have shown that it is possible to retain robustness while moving towards a more biological learning rule. The development of a clustered organization in the RNN allows a large drive for the read-out neurons while keeping the individual synaptic strengths reasonably small. If the clusters become small, larger read-out synaptic strengths are required, and the dynamics become less robust. Indeed, the sequential dynamics is especially fragile in the limit where every cluster has exactly one neuron. Furthermore, we show that learning is faster with more neurons per cluster since relatively small changes in the synapses are sufficient to learn the target. This is consistent with the intuitive idea that some redundancy in the network can lead to an increased learning speed [Raman et al., 2019]. In its current form, the target pattern needs to be presented repeatedly to the network and does not support one-shot learning through a single supervisor presentation. Although increasing the cluster size $N_{C}$ can reduce the number of presentations, the size of the clusters would need to be impractically large for one-shot learning. Alternatively, increasing the learning rate of the plastic read-out weights could be a way to reduce the number of target presentations to just one [Nicola and Clopath, 2019]. However, it is unclear at present whether such high learning rates are supported by experiments [Sjöström et al., 2001].

Taken together, our numerical simulations suggest ways to scale our network. An optimal network configuration can be chosen given the temporal length of the target sequence, and requirements on the temporal precision and robustness. For example, we have shown that a network with $N_{E}=6400$ and $N_{C}=200$ can be chosen for a $400$ m$s$ target sequence that can be learned fast with good temporal precision and a robust replay. If the network configuration and size are fixed, this severely constrains the sequences that can be learned and how they are replayed.

In this paper, we use local Hebbian learning to produce a sequential dynamics in the recurrent network. This is in contrast with previous studies, where often a recursive least squares method is used to train the weights of the recurrent network [Rajan et al., 2016, Hardy and Buonomano, 2018]. Hardcoding a weight structure into the recurrent network has been shown to result in a similar sequential dynamics [Chenkov et al., 2017, Setareh et al., 2018, Spreizer et al., 2019]. Studies that do incorporate realistic plasticity rules are mostly limited to purely feedforward synfire chains [Fiete et al., 2010, Waddington et al., 2012, Zheng and Triesch, 2014]. Those studies focus on the generation of a dynamics that is sequential. In this regard, the contribution of our work is to use the sequential dynamics as a key element to learning spatiotemporal spiking patterns. The ubiquity of sequential dynamics in various brain regions [Ikegaya et al., 2004, Jin et al., 2009, Mackevicius et al., 2019] and the architecture of the songbird system [Fee and Scharff, 2010] were an inspiration for the proposed separation of temporal and spatial information in our setup. As we have shown, this separation enables the use of a local Hebbian plasticity rule in the read-out synapses. Our model would therefore predict that perturbing the sequential activity should lead to learning impairments. Further perturbation experiments can test this idea and shed light on the mechanisms of sequential learning [Hemberger et al., 2019].

Previous studies have discussed whether sequences are learned and executed serially or hierarchically [Lashley, 1951]. Our recurrent network has a serial organization. When the sequential activity breaks down halfway, the remaining clusters are not activated further. A hierarchical structure would avoid such complete breakdowns at the cost of a more complicated hardware to control the system. Sequences that are chunked in sub-sequences can be learned separately and chained together. When there are errors in early sub-sequences this will less likely affect the later sub-sequences. A hierarchical organization might also improve the capacity of the network. In our proposed serial organization, the number of spatiotemporal patterns that can be stored is equal to the number of read-out neurons. A hierarchical system could be one way to extract general patterns and reduce the number of necessary read-out neurons. Evidence for hierarchical structures is found throughout the literature [Sakai et al., 2003, Glaze and Troyer, 2006, Okubo et al., 2015]. The basal ganglia is for example thought to play an important role in shaping and controlling action sequences [Tanji, 2001, Jin and Costa, 2015, Geddes et al., 2018]. Another reason why a hierarchical organization seems beneficial is inherent to the sequential dynamics. The time-variability of the sequential activity grows by approximately $\sqrt{t}$ (see Fig 5E)). While on a time scale of a few hundreds of milliseconds, this does not pose a problem, for longer target sequences this variability would exceed the plasticity time constants. The presented model could thus serve as an elementary building block of a more complex hierarchy.

In summary, we have demonstrated that a clustered network organization can be a powerful substrate for learning, moving biological learning systems closer to machine learning performance. Specifically, the model dissociates temporal and spatial information and therefore can make use of Hebbian learning to learn spatiotemporal sequences over behavioural time scales. More general, the backbone as a clustered connectivity might encode any variable $x$ and enable downstream read-out neurons to learn and compute any function of this variable, $\phi(x)$.

Methods

Neuron and synapse models

Excitatory neurons are modelled with the adaptive exponential integrate-and-fire model [Brette and Gerstner, 2005]. A classical integrate-and-fire model is used for the inhibitory neurons. All excitatory to excitatory recurrent synapses are plastic under the voltage-based STDP rule [Clopath et al., 2010]. This enables the creation of neuronal clusters and a feedforward structure. Normalization and weight bounds are used to introduce competition and keep the recurrent network stable. Synapses from inhibitory to excitatory neurons in the recurrent network are also plastic under a local plasticity rule [Vogels et al., 2011]. In general, it prevents runaway dynamics and allows for an automatic search of good parameters (Fig S5). The connections from the recurrent network to the read-out neurons are plastic under the same voltage-based STDP rule. However, potentiation of read-out synapses is linearly dependent on the strength of the synapses. There is no normalization here to allow a continuous weight distribution. The dynamics was chosen based on previous models, with parameters for the dynamics and plasticity to a large extent conserved [Litwin-Kumar and Doiron, 2014]. More simple integrate-and-fire dynamics should lead to the same qualitative results, given that the parameters are appropriately changed (data not shown).

Network dynamics

Recurrent network

A network with $N^{E}$ excitatory ($E$) and $N^{I}$ inhibitory ($I$) neurons is homogeneously recurrently connected with connection probability $p$. Our network is balanced in terms of inhibition and excitation, so that it displays irregular and asynchronous spiking. This is signalled by the coefficient of variation (CV) of the inter-spike intervals of the neurons being $CV\sim 1$, thus indicating Poisson-like spiking [Brunel, 2000]. In our construction, we initialise the weights of the network near the balanced state by scaling the weights of the balanced RNN in Ref. [Litwin-Kumar and Doiron, 2014] by the square root of the relative network size. We then verify that the scaled parameters indeed lead to irregular dynamics. The spiking dynamics is slightly more regular on average, with a mean $CV\sim 0.8$ for excitatory neurons and a mean $CV\sim 0.9$ for inhibitory neurons.

Read-out neurons

The $N^{E}$ excitatory neurons from the recurrent network are all-to-all connected to $N^{R}$ excitatory read-out ($R$) neurons. This weight matrix is denoted by $W^{RE}$ and it is where the learned sequence is stored. To help learning, there are two additional types of neurons in the read-out network. During learning, the read-out neurons receive supervisory input from $N^{R}$ excitatory supervisor ($S$) neurons. The connection from supervisor neurons to read-out neurons is one-to-one and fixed, $w^{RS}$. Also during learning, $N^{R}$ interneurons ($H$) are one-to-one and bidirectionally connected to the read-out neurons with fixed connection strengths, $w^{RH}$ and $w^{HR}$ (see Table 1 for the recurrent network and read-out parameters). The $E$ to $E$, $I$ to $E$ and the $E$ to $R$ connections are plastic.

Membrane potential dynamics

There are two different regimes, one for each part of the model. Excitatory neurons in the recurrent network have a high adaptation current while excitatory neurons in the read-out network have no adaptation. This is to allow for a wide range of firing rates in the read-out network, while spiking is more restricted in the recurrent network. Differences in the refractory period are there for the same reason, but are not crucial. The membrane potential of the excitatory neurons ($V^{E}$) in the recurrent network has the following dynamics:

[TABLE]

where $\tau^{E}$ is the membrane time constant, $E_{L}^{E}$ is the reversal potential, $\Delta_{T}^{E}$ is the slope of the exponential, $C$ is the capacitance, $g^{EE},g^{EI}$ are synaptic input from excitatory and inhibitory neurons respectively and $E^{E},E^{I}$ are the excitatory and inhibitory reversal potentials respectively. When the membrane potential diverges and exceeds $20$ mV, the neuron fires a spike and the membrane potential is reset to $V_{r}$. This reset potential is the same for all neurons in the model. There is an absolute refractory period of $\tau_{abs}$. The parameter $V_{T}^{E}$ is adaptive for excitatory neurons and set to $V_{T}+A_{T}$ after a spike, relaxing back to $V_{T}$ with time constant $\tau_{T}$:

[TABLE]

The adaptation current $a^{E}$ for recurrent excitatory neurons follows:

[TABLE]

where $\tau_{a}$ is the time constant for the adaptation current (see also Fig S6). The adaptation current is increased with a constant $\beta$ when the neuron spikes. The membrane potential of the read-out ($V^{R}$) neurons has no adaptation current:

[TABLE]

where $\tau^{E}$, $E_{L}^{E}$, $\Delta_{T}^{E}$, $E^{E}$, $E^{I}$ and $C$ are as defined before. $g^{RE}$ is the excitatory input from the recurrent network. $g^{RS}$ is the excitatory input from the supervisor neuron (supervisor input only non-zero during learning, when the target sequence is repeatedly presented). $g^{RH}$ is the inhibitory input from the interneuron (only non-zero during learning, to have a gradual learning in the read-out synapses). The absolute refractory period is $\tau_{absR}$. The threshold $V_{T}^{R}$ follows the same dynamics as $V_{T}^{E}$, with the same parameters. The membrane potential of the supervisor neurons ($V^{S}$) has no inhibitory input and no adaptation current:

[TABLE]

where the constant parameters are defined as before and $g^{SE}$ is the external excitatory input from the target sequence. The absolute refractory period is $\tau_{absS}$. The threshold $V_{T}^{S}$ follows again the same dynamics as $V_{T}^{E}$, with the same parameters. The membrane potential of the inhibitory neurons ($V^{I}$) in the recurrent network has the following dynamics:

[TABLE]

where $\tau^{I}$ is the inhibitory membrane time constant, $E_{L}^{I}$ is the inhibitory reversal potential and $E^{E},E^{I}$ are the excitatory and inhibitory resting potentials respectively. $g^{EE}$ and $g^{EI}$ are synaptic input from recurrent excitatory and inhibitory neurons respectively. Inhibitory neurons spike when the membrane potential crosses the threshold $V_{T}$, which is non-adaptive. After this, there is an absolute refractory period of $\tau_{abs}$. There is no adaptation current. The membrane potential of the interneurons ($V^{H}$) follow the same dynamics and has the same parameters, but there is no inhibitory input:

[TABLE]

where the excitatory input $g^{HE}$ comes from both the read-out neuron it is attached to and external input. After the threshold $V_{T}$ is crossed, the interneuron spikes and an absolute refractory period of $\tau_{absH}$ follows. The interneurons inhibit the read-out neurons stronger when they receive strong inputs from the read-out neurons. This slows the potentiation of the read-out synapses down and keeps the synapses from potentiating exponentially (see Table 2 for the parameters of the membrane dynamics).

Synaptic dynamics

The synaptic conductance of a neuron $i$ is time dependent, it is a convolution of a kernel with the total input to the neuron $i$:

[TABLE]

where $X$ and $Y$ denote two different neuron types in the model ($E$, $I$, $R$, $S$ or $H$). $K$ is the difference of exponentials kernel:

[TABLE]

with a decay time $\tau_{d}$ and a rise time $\tau_{r}$ dependent only on whether the neuron is excitatory or inhibitory. There is no external inhibitory input to the supervisor and inter- neurons. During spontaneous activity, there is no external inhibitory input to the recurrent network and a fixed rate. The external input to the interneurons has a fixed rate during learning as well. The external input to the supervisor neurons is dependent on the specific learning task. There is no external input to the read-out neurons. The externally incoming spike trains $s_{ext}^{X}$ are generated from a Poisson process with rates $r_{ext}^{X}$. The externally generated spike trains enter the network through synapses $W_{ext}^{X}$ (see Table 3 for the parameters of the synaptic dynamics).

Plasticity

Excitatory plasticity

The voltage-based STDP rule is used [Clopath et al., 2010]. The synaptic weight from excitatory neuron $j$ to excitatory neuron $i$ is changed according to the following differential equation:

[TABLE]

Here, $A_{LTD}$ and $A_{LTP}$ are the amplitude of depression and potentiation respectively. $\theta_{LTD}$ and $\theta_{LTP}$ are the voltage thresholds to recruit depression and potentiation respectively, as $R(.)$ denotes the linear-rectifying function ($R(x)=0$ if $x<0$ and else $R(x)=x$). $V_{i}$ is the postsynaptic membrane potential, $u_{i}$ and $v_{i}$ are low-pass filtered versions of $V_{i}$, with respectively time constants $\tau_{u}$ and $\tau_{v}$ (see also Fig S6):

[TABLE]

where $s_{j}$ is the presynaptic spike train and $x_{j}$ is the low-pass filtered version of $s_{j}$ with time constant $\tau_{x}$:

[TABLE]

Here the time constant $\tau_{x}$ is dependent on whether learning happens inside ($E$ to $E$) or outside ($E$ to $R$) the recurrent network. $s_{j}(t)=1$ if neuron $j$ spikes at time $t$ and zero otherwise. Competition between synapses in the recurrent network is enforced by a hard $L1$ normalization every $20$ m$s$, keeping the sum of all weights onto a neuron constant: $\sum_{j}W_{ij}=K$. $E$ to $E$ weights have a lower and upper bound $[W_{min}^{EE},W_{max}^{EE}]$. The minimum and maximum strengths are important parameters and determine the position of the dominant eigenvalues of $W$. Potentiation of the read-out synapses is weight dependent. Assuming that stronger synapses are harder to potentiate [Debanne et al., 1999], $A_{LTP}$ reduces linearly with $W^{RE}$:

[TABLE]

The maximum LTP amplitude $A$ is reached when $W^{RE}=W_{min}^{RE}$ (see Table 4 for the parameters of the excitatory plasticity rule).

Inhibitory plasticity

Inhibitory plasticity acts as a homeostatic mechanism, previously shown to prevent runaway dynamics [Vogels et al., 2011, Litwin-Kumar and Doiron, 2014, Zenke et al., 2015]. Here, it allows to automatically find good parameters (see also Fig S5). Excitatory neurons that fire with a higher frequency will receive more inhibition. The $I$ to $E$ weights are changed when the presynaptic inhibitory neuron or the postsynaptic excitatory neuron fires [Vogels et al., 2011]:

[TABLE]

where $r_{0}$ is a constant target rate for the postsynaptic excitatory neuron. $s^{E}$ and $s^{I}$ are the spike trains of the postsynaptic $E$ and presynaptic $I$ neuron respectively. The spike trains are low pass filtered with time constant $\tau_{y}$ to obtain $y^{E}$ and $y^{I}$ (as in equation 12). Table 5 shows parameter values for the inhibitory plasticity rule. The $I$ to $E$ synapses have a lower and upper bound $[W_{min}^{IE},W_{max}^{IE}]$.

Learning protocol

Learning happens in two stages. First a sequential dynamics is learned in the RNN. Once this temporal backbone is established connections to read-out neurons can be learned. Read-out neurons are not interconnected and can learn in parallel.

Recurrent network

The network is divided in $30$ disjoint clusters of $80$ neurons. The clusters are sequentially stimulated for a time duration of $60$ minutes by a large external current where externally incoming spikes are drawn from a Poisson process with rate $18$ k$Hz$. This high input rate does not originate from a single external neuron but rather assumes a large external input population. Each cluster is stimulated for $10$ m$s$ and in between cluster stimulations there are $5$ m$s$ gaps (see also Fig S6 for different gaps). During excitatory stimulation of a cluster, all other clusters receive an external inhibitory input with rate $4.5$ k$Hz$ and external input weight $W_{ext}^{I}=2.4$ p$F$. There is a periodic boundary condition, i.e. after the last cluster is activated, the first cluster is activated again. After the sequential stimulation, the network is spontaneously active for $60$ minutes. The connectivity stabilizes during the spontaneous dynamics. Learning in scaled versions of this network happens in exactly the same way (Fig 5A). The recurrent weight matrix of the large network ($80$ clusters of $80$ neurons, Figs 5E and 6) is learned using the same protocol. The recurrent weight matrix reaches a stable structure after three hours of sequential stimulation followed by three hours of spontaneous dynamics. Parameters that change for the scaled up version are summarized in Table 6. For randomly switching dynamics, a similar protocol is followed (Fig S1). The weight matrix used to plot the spectrum of the recurrent network in Figs 2 and S2 is:

[TABLE]

Read-out network

During learning of the read-out synapses, external input drives the supervisor and inter- neurons. The rate of the external Poisson input to the supervisor neurons reflects the sequence that has to be learned. The rate is normalized to be between [math] k$Hz$ and $10$ k$Hz$. During learning, $W^{RE}$ changes. After learning, the external input to the supervisor and inter- neurons is turned off and both stop firing. The read-out neurons are now solely driven by the recurrent network. Plasticity is frozen in the read-out synapses after learning. With plasticity on during spontaneous dynamics, the read-out synapses would continue to potentiate because of the coactivation of clusters in the recurrent network and read-out neurons. This would lead to read-out synapses that are all saturated at the upper weight bound.

Simulations

The code used for the training and simulation of the recurrent network is built on top of the code from [Litwin-Kumar and Doiron, 2014] in Julia. The code used for learning spatiotemporal sequences using read-out neurons is written in Matlab. Forward Euler discretization with a time step of $0.1$ m$s$ is used. The code is available for the reviewers on ModelDB (http://modeldb.yale.edu/257609) with access code ’SSRSNT’. The code will be made public after publication.

Linear rate model: spectral analysis

A linear rate model can give insight into the dynamics of a large nonlinear structured spiking network [Schaub et al., 2015]. The dynamics of a simplified rate model with the same feedforward structure as in the RNN is as follows:

[TABLE]

where $x$ is a multidimensional variable consisting of the rates of all excitatory and inhibitory clusters, $A$ is the weight matrix, and $\xi$ is white noise. The matrix $A$ is a coarse-grained version of the weight matrix of the recurrent network in Fig 2B averaged over each cluster. In order to obtain analytical expressions, we consider a network with $3$ excitatory clusters and 1 inhibitory cluster. The connectivity of this model can be parametrized as:

[TABLE]

where $\delta>0$, $w=\delta+\epsilon+1$, $\epsilon>1$ guarantees sequential dynamics, and $k>1$ guarantees a balanced network.

The Schur decomposition $A=UTU^{T}$ gives eigenvalues and Schur vectors $\mathbf{u}_{i}$:

[TABLE]

[TABLE]

The first mode $\mathbf{u}_{1}$ decays fast and uniformly over the four neuronal groups. The second mode $\mathbf{u}_{2}$ decays more slowly, and indicates the interplay between excitatory groups $(x_{1},x_{2},x_{3})$ and the inhibitory group $x_{4}$. The eigenspace associated with the pair $\{\mathbf{u}_{3},\mathbf{u}_{4}\}$ has complex conjugate eigenvalues and is localized on the three excitatory groups. An increase of activity in one excitatory group is coupled with decreased activities in the other groups. If the real part $\delta-\frac{\epsilon+1}{2}<1$ then these modes are linearly stable but if the real part becomes closer to one means a slower decay of this mode. Importantly, the imaginary part of the eigenvalues is $\pm\sqrt{3}(\epsilon-1)/2$; hence it grows linearly with the strength of the feedforward structure $(\epsilon-1)$ (Fig 7A). This leads to oscillatory behavior, which determines the time scale of the sequential switching.

Supplementary material