Autonomous emergence of global structure

Synfire chains, consisting of distinct groups of neurons that project to each other in a sequential order (Fig 1A), were originally proposed as a model for sequence generation in the mammalian cortex [20]. Subsequently, compelling evidence has pointed to the possibility that this architecture underlies the synchronous neural activity observed in the songbird premotor nucleus HVC [21, 22], with ∼100 excitatory neurons in each layer. Theoretical reasoning indicates that this architecture can indeed produce stable propagation of synchronous spiking activity if the groups are sufficiently large [23, 24].

It was shown theoretically that STDP, combined with heterosynaptic competition, can lead spontaneously to the formation of thin chains [25], in which single neurons project to each other sequentially (see also [26]). Formation of wide synfire chains, in which many neurons participate in each group along the sequence, proved to be more difficult, and was successfully demonstrated when structured inputs were fed into the network [25]. Thus it remains unclear whether structured inputs are required for the formation of wide chains in networks of spiking neurons. Here we demonstrate that with appropriate choice of the biophysical parameters, STDP dynamics can lead to robust formation of wide synfire chains, without the need to provide any structured inputs to the network. The grouping of neurons occurs spontaneously, assisted by high order synaptic interactions that arise from the STDP dynamics.

As a second example for the role of high order synaptic interactions in plasticity, we consider whether STDP can promote the formation of distinct, self connected groups of cells (Fig 1B). Theoretical works demonstrated that such structures can lead to multiple persistent states in the neural dynamics [27–31]. Another motivation for consideration of these structures arises from anatomical studies of local connectivity: for example, connections among excitatory neurons in the rat visual and somatosensory cortices tend to be clustered [1, 32]. In the context of learning, it has been demonstrated theoretically that STDP, combined with additional plasticity mechanisms and structured inputs, can lead to formation of self connected assemblies [27, 33, 34]. However, in similarity to the formation of synfire chains by STDP, it remains unclear whether self connected assemblies can emerge in an initially unstructured network without the inclusion of correlated inputs that are fed into the network during learning. Here we show that high order synaptic interactions enable the spontaneous formation of self connected assemblies, without the inclusion of such inputs.

Synaptic drift in recurrent networks

In studying the effects of STDP, it is necessary to consider models of neural activity that explicitly involve the timing of action potentials (as opposed to simpler rate models). Due to the difficulty in evaluating spike correlation functions in most models of spiking neurons, analytical treatments of STDP have made certain approximations for the spiking statistics: typically, pre and post synaptic spike trains were treated as if they follow inhomogeneous Poisson statistics [12–14, 16]. Therefore, we explicitly consider a recurrent network of neurons which follow linear Poisson (LP) dynamics (Methods). The activity in the network is stochastic, and the probability of each neuron to emit an action potential is proportional to a weighted sum of the previous activity in the network and a constant external input (see Fig 2A and Methods).

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Fig 2.

A. Illustration of the network architecture. The network contains recurrently connected neurons that receive a constant external input. B. Example of an STDP function. Horizontal axis: time interval between action potentials in the pre and post synaptic neurons. Vertical axis: change in synaptic efficacy. C. The drift of the STDP dynamics is an integral over the product of the correlation and STDP functions. Black line: time dependent correlation. Dashed line: STDP function. The area under the product of these curves (yellow) determines the synaptic drift.

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Networks of LP neurons have been shown to approximate well the correlation in spike timing of neurons with more elaborate leaky integrate-and-fire dynamics, operating in an asynchronous regime [35]. The availability of an exact expression for spike correlations in such networks (see below) allows us to develop a precise theory for the weight dynamics driven by STDP.

We consider synaptic efficacies in a recurrent neural network with an arbitrary structure (Fig 2A). The efficacies undergo long term potentiation or depression in response to each pair of spikes, depending on the time interval between the firing of the pre and post synaptic neurons (Fig 2B). In the case of slow learning rate, the rate of change in the synaptic efficacies can be expressed in terms of the product between the time dependent pair correlation and the STDP function (see Methods): (1) Here, is the drift in the synaptic efficacy from neuron j to neuron i, defined as the average change in the synaptic efficacy per unit time. The correlation function C_ij(τ) represents the probability density that neurons j and i emit a pair of spikes temporally separated by a duration τ (Eq 9), and F(τ) is the STDP function that describes how potentiation (or depression) depends on the time interval τ (Fig 2C).

Using an analytic expression for the spike correlation function [36], we obtain an exact expression for STDP in recurrent networks with arbitrary connectivity (Methods): (2) Here, W is the connectivity matrix, a(t) is the time course of synaptic currents, is its Fourier transform, r_i is the average firing rate of neuron i (Eq 13), D_ij = δ_ij r_j, and f₀ is the area under the STDP function (Eq 16). The derivation of Eq 2 does not involve any assumptions on the specific form of the synaptic currents, STDP function, or the network architecture. S1 Fig demonstrates that this expression provides an accurate description of the average learning dynamics in networks of LP neurons, over time scales which are relevant to plasticity in the brain.

The synaptic drift can be interpreted as a sum over contributions from structural motifs

Eq 2 expresses how plasticity in one synapse depends on the connectivity of the full network. Additional insight on this expression, which may seem elaborate, is obtained by noticing that the spike correlation functions in the network can be written as a power series, obtained as an expansion in the strength of the synaptic efficacies [37]. This allows us to reformulate Eq 2 as follows (Methods): (3) where the coefficients f_{α, β} are defined below. Each one of the terms in Eq 3 has a relatively simple, intuitive interpretation that we discuss next.

The first term in Eq 3, f₀ r_i r_j, represents the contribution to STDP arising from the mean firing rates of the pre and post synaptic neurons, while ignoring any correlations in the timing of their spikes. Accordingly, this term is simply proportional to the firing rates r_i and r_j (Fig 3A). Such a term is often postulated in phenomenological models of synaptic plasticity [38], and its emergence from STDP dynamics has been described, e.g., in [11, 12, 16, 17].

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Fig 3. Network motifs that affect the STDP dynamics.

A. The first term in Eq 3 is proportional to the average firing rates of the pre and post synaptic neurons. B. General form of a structural motif, contributing to the STDP dynamics in the synapse W_ij (dashed line). A source neuron k (red line) projects to the post synaptic neuron i (green) via α synapses, and to the pre synaptic neuron j (black) via β synapses. The contribution is proportional to the firing rate of the source neuron r_k and to the product of all synaptic efficacies along the two paths. The expression (W^α)_ik r_k(W^β)_jk is a sum over all paths that start from neuron k and reach the post synaptic neuron i via α synapses and the pre synaptic neuron j via β synapses. C-H: Examples of six motifs that affect STDP in the synapse W_ij (dashed line), by inducing time dependent coupling between neurons j and i. Black arrows indicate synapses that participate in the motifs. C. One of the first order motifs contains the direct synapse from j to i (α = 1, β = 0). The source neuron is the pre synaptic neuron j. D. The other first order motif contains the opposite synapse from i to j (α = 0, β = 1). Here the source neuron is the post synaptic neuron i. E. One of the second order motifs (α = 1, β = 1). Here the source neuron is some other neuron in the network that projects directly both to i and j. F. Additional example for a second order motif (α = 2, β = 0). The source neuron is the pre synaptic neuron j, that projects to the post synaptic neuron i through one intermediate neuron. G. Example of a third order motif (α = 2, β = 1). The source neuron projects directly to the pre synaptic neuron j and via one intermediate neuron to the post synaptic neuron i. H. Additional example of a third order motif (α = 3, β = 0). The pre synaptic neuron j is the source neuron, that projects to the post synaptic neuron i via two intermediate neurons.

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The probability of neurons i and j to emit a spike is transiently modulated whenever a spike is emitted anywhere within the network. In the sum on the right hand side of Eq 3, each term quantifies how a spike in a neuron k modulates the probability of neurons i and j to emit pairs of spikes at various latencies—and through this modulation, how the spikes of neuron k influence the drift in the synaptic efficacy W_ij. This contribution to the drift is written as a sum over structural motifs, which share a common organization shown schematically in Fig 3B.

In each structural motif, the source neuron k projects to the post synaptic neuron i via a path of α synapses, and to the pre synaptic neuron j via a path of β synapses. The synaptic drift driven by the motif is proportional to all the synaptic weights along the two paths, and to the firing rate of the source neuron. In addition, the drift is proportional to a motif coefficient f_{α, β}. This coefficient depends on the number of synapses in the two paths, the time course of the synaptic currents, and the detailed form of the STDP learning function. We discuss this dependence in detail later (see also Methods).

The first order contributions in the above sum are those in which {α, β} = {1, 0} (Fig 3C), or {α, β} = {0, 1} (Fig 3D): (4) These terms are local: they depend only on the direct synapses that link neurons i and j, and on the firing rate of these two neurons (Fig 3C and 3D). Previous works [17, 18] derived these contributions to STDP using heuristic arguments that focused on the pre and post synaptic neurons, and studied their consequences when embedded in a recurrent network. Under an asymmetric STDP function, the first order terms induce a competition between a synapse W_ij and the opposite synapse W_ji [17, 18], whereas a symmetric STDP function tends to promote the development of a symmetric weight matrix. Here, these local plasticity rules are obtained as the first order terms in a systematic expansion, which includes also higher order terms.

High order motifs can promote self organization into global structures

Next, we demonstrate that high order motifs can promote the formation of large-scale structures in the synaptic connectivity. We focus on two types of structures: synfire chains (Fig 4A), and clusters of self connected assemblies (Fig 4B). In both structures, synapses of different neurons are highly correlated. The purpose of this section is to illustrate by specific examples that high order motifs, beyond the first order, can lead to emergence of these structural correlations. A more systematic study is presented in later sections.

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Fig 4. Self organization into global structures.

A. The third order motif {2, 1} promotes formation of wide synfire chain connectivity, by encouraging neurons that receive input from the same pre synaptic neuron to project to the same post synaptic neuron. B. The second order motif {1, 1}, combined with a symmetric STDP function, promotes formation of self connected assemblies by encouraging two neurons that receive inputs from a common source to increase their synaptic coupling reciprocally. C-D. Steady state connectivity, obtained from simulations in which the motif coefficients f_{α, β} are tuned artificially. STDP is combined with heterosynaptic competition. Neurons are ordered such that the matrix represents the optimal feed forward connectivity (C) or self connected assemblies connectivity (D). C. Left—results from dynamics that contain only the motifs {1, 0}, {0, 1}, {2, 0}, {0, 2}. The simulation does not converge into synfire chains connectivity. Right—the simulation contains also the motifs {2, 1}, {1, 2}. The simulation converges into synfire chain connectivity. D. Left—the dynamics contain only the motifs {1, 0}, {0, 1}. The simulation does not converge to assemblies. Right—the dynamics include also the motif {1, 1}. The connectivity converges into self connected assemblies. E. The drift in the excitatory connections is driven by three plasticity mechanisms: (i) Spike timing dependent plasticity. (ii) Heterosynaptic competition between all synapses that terminate in the same post synaptic neuron (red arrows), and all synapses that originate from the same pre synaptic neuron (green arrows). (iii) A hard bound on the synaptic efficacy.

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We consider networks that consist of recurrently connected excitatory neurons, and inhibitory interneurons with fast synapses (see Methods). The inhibitory input to each neuron depends on the total activity of the excitatory network, and is adjusted such that the inhibitory weights balance the excitatory ones [14, 15]. The main role of inhibition in the network is to suppress runaway excitation in the neural dynamics. In addition, all neurons receive constant, identical inputs (Methods).

The plasticity mechanisms acting on the excitatory neurons are summarized in Fig 4E. The excitatory connections are modifiable through STDP and are bounded between zero and a positive bound, denoted by w_max. In addition to STDP, the excitatory synapses undergo heterosynaptic competition that limits the total synaptic input and output of each neuron: the sum of the incoming excitatory weights to each neuron, and the sum of outgoing excitatory weights from each neuron are bounded to lie below a positive hard bound, denoted by W_max. The competition, combined with STDP and with the hard bound on each synaptic weight, can lead to a steady state connectivity pattern in which each neuron receives input from a certain number of pre-synaptic partners, and projects to a certain number of post-synaptic partners [25]. These numbers are tuned by the ratio between W_max and w_max [25].

To test the influence of individual motifs on the STDP dynamics, we perform simulations in which we include only a few of the terms in Eq 3, starting from initial weights that were drawn independently from a uniform random distribution (Methods). Instead of obtaining the exact expressions for the coefficients f_{α, β} (Eq 15), we artificially tune their values and observe the consequences on the structures that emerge.

The motif {2, 1} (Fig 3G), when acting between excitatory neurons, encourages neurons that receive input from the same pre synaptic neuron to project into the same post synaptic neuron (Fig 4A). Consequently, this motif can induce correlations between the synaptic connections formed by neurons that belong to the same layer of a synfire chain. This is a key feature which differentiates wide synfire chain structures from other connectivity patterns in which each neuron has a prescribed number of presynaptic and postsynaptic partners. This observation raises a hypothesis, that the motif {2, 1} can promote formation of wide synfire chains, by favoring these structures over other connectivity patterns which are compatible with the constraints set by the heterosynaptic competition.

To test this hypothesis, we perform simulations that include only contributions from the motifs {α, β} = {1, 0}, {2, 0}, {2, 1}, and the opposing terms in which α and β are exchanged. Additionally, we set f_{α, β} = −f_{β, α}, as expected if the STDP function is antisymmetric.

An example is shown in Fig 4C. When including only the first and second order motifs, a simulation of the plasticity dynamics leads to a structure in which each row and column contains a small number of active synapses, without any reciprocal connections (left). However, the synaptic weights are not organized in a synfire chain structure. When including the third order motifs {2, 1} and {1, 2}, the synaptic efficacies self organize into a perfect synfire chain (right), despite the absence of any correlations in the external inputs to the network. Results from a wider set of simulations, in which we systematically vary the strength of motifs, are presented in the section Self organization into synfire chains.

As a second example, we examine the influence of the motif {1, 1} on the formation of self connected cell assemblies. The motif {1, 1} (Fig 3E), when acting between excitatory neurons, enhances reciprocal connections between neurons that receive common input (note that the contribution from this motif vanishes if the STDP function is antisymmetric, but for a symmetric STDP function this motif can significantly contribute to the plasticity dynamics). This raises the hypothesis that formation of self connected assemblies can be promoted by the contribution of the motif {1, 1}.

To check this hypothesis, we conduct plasticity simulations that contain only contributions from first and second order motifs: {0, 1}, {1, 0}, and {1, 1} in Eq 3. In addition, we impose the relation f_1,0 = f_0,1, as expected if the STDP function is symmetric (see Eq 15 in Methods). In the example shown in Fig 4D, the first order motifs, together with the synaptic competition, lead to a symmetric connectivity matrix in which the number of active synapses is limited in each row and in each column (left). With the inclusion of the {1,1} motif (right), strong correlations emerge between synapses of different neurons, and fully connected cell assemblies emerge.

Biophysical properties affect the relative contribution of different motifs to STDP

So far, we illustrated how high order motifs can promote the formation of global structures by artificially tuning the contribution of specific motifs to plasticity (through the coefficients f_{α, β}). In the actual STDP dynamics, the coefficients f_{α, β} cannot be controlled independently. Instead, these coefficients are determined by the temporal structure of the STDP function and the synaptic currents.

Each motif induces temporal correlations of spikes between the pre and post synaptic neurons with a characteristic time course. The time course of correlations depends on the structure of the motif (characterized by α and β), and on the time course of the synaptic currents. Therefore, the synaptic current, together with the STDP function, affects how strongly each motif influences the synaptic drift, as quantified by the magnitude of the corresponding motif coefficient f_{α, β} (Eq 15). The influence of the synaptic currents on the motif coefficients is illustrated in detail in Methods.

As a specific example for how the time course of synaptic currents can affect the motif coefficients, we consider in Fig 5 the influence of a delay in the onset of the post synaptic current (abbreviated below as the synaptic latency). Synaptic latencies depend on diverse physiological properties, such as the length and conductance velocity in the axon [39] and location on the dendrite [40]. Here we model the synaptic latency as a temporal shift in the synaptic current (Fig 5).

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Fig 5.

Left: Motif coefficients as a function of the synaptic latency. Solid line—f_1,0, dashed line—f_2,0, dotted line—f_2,1. An antisymmetric STDP function is assumed. Right: the time course of the synaptic current. The synaptic latency is modeled as a delay in the onset of the postsynaptic current, relative to the presynaptic spike time. The precise form of the synaptic current and STDP function are specified in Methods.

https://doi.org/10.1371/journal.pcbi.1005056.g005

In the example shown in Fig 5 the motif coefficients f_1,0 and f_2,0 decrease with an increase of the synaptic latency. The coefficient f_2,1 is influenced by the synaptic latency as well, but for synaptic latencies ranging from 0 to 10 ms this dependence is extremely weak. Therefore, an increase in the synaptic latency reduces the contribution of the motifs {1, 0} and {2, 0} to STDP relative to the motif {2, 1}. The reason for these trends is explained qualitatively in Methods. Higher order motifs exhibit a similar behavior, depending on the difference between α and β (S5 Fig).

The possibility to tune the relative contribution of motifs through the interplay of post synaptic currents and the STDP function, suggests that global structures could be spontaneously generated via STDP with appropriate choice of these biophysical parameters. Furthermore, this observation provides a principled way to search for parameters that enable emergence of specific structures.

Self organization into synfire chains

We next focus on the emergence of synfire chains under the influence of STDP and heterosynaptic competition. For simplicity, we consider mainly the case where the STDP function is antisymmetric.

First, we consider in detail the interplay between the third order motif {2, 1}, which facilitates the formation of synfire chains, and the first order motifs {1, 0} and {0, 1}, whose contribution to STDP dynamics was the focus of previous theoretical works [17, 18]. Intrinsic plasticity mechanisms other than STDP can act to self-regulate the efficacy of each synapse, in similarity to the effect of the first order motif {1, 0} (see below). Therefore, it is interesting to consider f_1,0 and f_0,1 as separate parameters even if the STDP function is antisymmetric. In Fig 6 we examine the phase space spanned by f_1,0, f_0,1, and f_2,1, while assuming that f_1,2 = −f_2,1 due to the antisymmetric form of the STDP function. To avoid decay of all weights to zero when f_1,0 is strongly negative, we include in the dynamics also a term that drives growth of each weight at a fixed rate (see Methods, Eq 19). For simplicity, we first consider a situation in which other motifs do not contribute to the dynamics.

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Fig 6. Chain score of the steady state connectivity, obtained from simulations that include selected motifs.

A. Density plot of the chain score, displayed as a function of the motif coefficients f_1,0 (horizontal axis) and f_0,1 (vertical axis). All other motif coefficients are kept fixed. Note that here f_2,1 = 20 (designated by a dotted line in panel B). Above: examples of steady state connectivity matrices for f_0,1 = −20, and (from left to right) f_1,0 = −30, 0, 30. B. Chain score as a function of the motif coefficients f_1,0 (horizontal axis) and f_2,1 (vertical axis) with all other motif coefficients kept fixed. Here f_0,1 = −30 (designated by a dotted line in panel A). Each data point represents an average over ten simulations, each with a different realization of the initial random connectivity. Additional details of the plasticity dynamics are specified in Methods.

https://doi.org/10.1371/journal.pcbi.1005056.g006

Fig 6A shows results from simulations in which we varied f_0,1 and f_1,0 while fixing f_2,1. To quantify whether synfire chains emerge robustly we constructed a score that quantifies similarity between the steady state excitatory connectivity and a perfect synfire chain structure (abbreviated below as the chain score). This chain score ranges from 0 to 1, where 1 corresponds to a perfect match (Methods). The figure shows the chain score, averaged over multiple random choices of the initial weights. Over a wide range of values of f_1,0 and f_0,1 precise synfire chain structures are obtained robustly. We note two characteristics of this parameter regime: first, the coefficient f_0,1 is negative. Thus, each synapse inhibits its reciprocal synapse. Second, f_1,0 lies within a range of values which is fairly insensitive to f_0,1 when |f_0,1| is sufficiently large.

Similarly, in Fig 6B we fix f_0,1, while varying f_1,0 and f_2,1. As expected, synfire chains emerge only when f_2,1 is sufficiently large. Furthermore, with increase of f_2,1 the range of values of f_1,0 that permits formation of synfire chains becomes wider. Thus, the high order motif {2, 1} plays a pivotal role in the spontaneous emergence of wide synfire chains. Results from additional simulations, in which we include additional low order motifs are shown in S2 Fig. Under typical conditions relevant to the full plasticity dynamics (discussed below), the second order motifs {2, 0} and {1, 1} have a detrimental effect on synfire chain formation (a contribution from the motif {1, 1} may be present if the STDP function is not antisymmetric). Next, we address the emergence of synfire chains under the full STDP dynamics.

Synaptic self-depression enables self-organization into synfire chains in the full STDP dynamics.

With typical choices of the STDP function and the synaptic currents, the contribution of the first order motif {1, 0} to the dynamics is large relative to higher order motifs (Fig 5). Consequently, based on the results shown in Fig 6 we do not expect emergence of synfire chain structures under STDP dynamics alone. To enable formation of synfire chains, we include in the dynamics a term that describes constant self depression of each excitatory synapse, which acts alongside STDP (see Methods). Such a term can arise, for example, from a self-regulating process which decreases the size of each dendritic spine in proportion to its volume [41]. Hence, we assume that the self depression term, acting on each synapse is proportional to the synaptic efficacy (− μW_ij in Eq 19). This form of synaptic self-depression selectively suppresses the contribution of the motif {1, 0}, since its contribution is similar to that of the motif {1, 0}. An increase in the rate of self-depression corresponds in Fig 6A to motion from right to left, in parallel to the horizontal axis. In light of Fig 6 we expect synfire chains to emerge for appropriate rates of the synaptic self-depression.

The black trace in Fig 7 shows results from simulations of the full STDP dynamics (Eq 2) combined with synaptic self-depression and heterosynaptic competition (Eq 19). All the processes contributing to the synaptic dynamics are summarized schematically in Fig 7A. In Fig 7B chain scores of the steady state connectivity, averaged over multiple simulations with random initial synaptic weights, are shown as a function of μ, the rate of self depression. All other parameters in the simulation are kept fixed. As expected, synfire chains emerge robustly when μ lies within an appropriate range.

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Fig 7. Emergence of synfire chains under the full STDP dynamics.

A. Schematic representation of the plasticity rules applied to excitatory synapses. In addition to the mechanisms mentioned in Fig 4, the synapses undergo self depression that weakens their efficacy in proportion to its magnitude. B-C. Chain score of the steady state connectivity, averaged over ten random choices of the initial connectivity. Black traces: complete dynamics (Eq 2); red (gray) traces—power series expansion (Eq 3), truncated to include motifs up to third (second) order. B. Chain score plotted as a function of μ, the relative rate of self-depression (the rate of self depression is given by ημ, where η is the learning rate, see Methods). All other parameters are kept fixed. C. Chain score plotted as a function of the synaptic latency, while keeping all other parameters fixed. Above: three examples of the steady state connectivity obtained in specific simulations, with different values of synaptic latency. Error bars in panels B-C represent the standard deviation of the chain score over the ten initial conditions.

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The red trace in Fig 7B shows similar results from simulations in which the expansion of the STDP dynamics (Eq 3) was truncated to include only contributions of motifs up to third order. Comparison with results of the full STDP dynamics (black line) indicates that the contributions of motifs up to third order are sufficient to predict quite well the conditions in which synfire chains emerge. In contrast, synfire chain structures do not emerge when the expansion in Eq 3 is truncated to include only motifs up to second order (Fig 7B, gray trace).

Robustness to noise.

The average synaptic drift as expressed by Eq 2 is sufficient to describe the plasticity dynamics when the learning rate is small and noise in the STDP dynamics, arising from random fluctuations in the number of pre and post synaptic spike pairs is averaged out. Thus, the analysis in the limit of slow learning does not guarantee that the network will exhibit similar plasticity dynamics in a more realistic scenario, where learning occurs over a biologically relevant time scale.

A rough estimate for the dependence of noise on the time scale of averaging can be obtained by comparing the prediction of the deterministic theory (Eq 2) with the synaptic efficacy generated in a stochastic spiking simulation. Such results are shown in S1 Fig, where the network parameters were chosen to roughly match those used in Fig 7. The synaptic drift is correctly predicted by Eq 2 and, as expected, the scatter relative to the predicted drift decreases with increase of the duration of averaging T. When averaged over a fairly short time scale of T = 6 min, the stochastic drift is strongly correlated with the prediction of the deterministic theory, but the scatter relative to the mean is fairly large (S1A Fig). When the duration of averaging is 200 times longer (T = 20 hours) the scatter is much smaller, and the agreement with Eq 2 is excellent (S1C Fig).

These results demonstrate that the deterministic theory provides a relevant prediction for the stochastic synaptic drift when the averaging is performed over a time scale of several minutes or more. However, during synfire chain formation the synaptic weights are constantly changing. Therefore, the whole process should occur over a significantly longer duration in order for the deterministic theory to be adequate. In Fig 8A–8C we examine the plasticity dynamics in neural networks with initial random synaptic weights, under parameters that enable robust emergence of synfire chain structures. The chain score is measured in stochastic simulations with varying values of the learning rate η, and compared with the prediction of the deterministic theory. Each data point in the figure represents an average of the chain score over ten randomly chosen initial conditions.

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Fig 8. Formation of synfire chains under stochastic dynamics.

A-B. Evolution in time of the chain score in the stochastic dynamics (red), compared with the prediction of the deterministic theory (black), shown for two different learning rates: η = 2 × 10⁻⁷ (A) and η = 1 × 10⁻⁷ (B) (see Methods for all other parameters). Solid traces: average over ten initial conditions. Limits of the shaded areas represent the standard deviation. Note the different time scales in A and B. C. Evolution in time of the chain score, shown for different learning rates (η) as a function of the time multiplied by η/η₀ where η₀ = 4 × 10⁻⁷. Each trace represents an average over ten random intial conditions, with a specific learning rate: η = 10⁻⁷ (red), 2 × 10⁻⁷ (green), 4 × 10⁻⁷ (blue), 6 × 10⁻⁷ (purple), and 10 × 10⁻⁷ (yellow). The black trace represents the prediction of the deterministic theory. Note that the traces for the smallest learning rates (green and red traces, corresponding to panels A and B, respectively) nearly collapse on a single line, in close agreement with the prediction of the deterministic theory. The legend shows the actual time corresponding to the right end of the horizontal axis for each one of the plots. The synaptic latency d = 6 ms. D. Chain score evaluated in simulations of the stochastic dynamics, shown as a function of the synaptic latency (the red circle corresponds to the synaptic latency in panel C). The learning rate η = η₀ = 4 × 10⁻⁷, matching the blue trace in panel C. Each data point represents an average over ten simulations with random initial conditions, and the error bars represent the standard deviation. The chain score is evaluated after 60 hours of simulated time.

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Within the deterministic theory the convergence time simply scales in proportion to η⁻¹. In contrast, in the stochastic simulations an increase in η entails an increase in the amplitude of noise (relative to the mean synaptic drift) and thus we can expect the agreement with the deterministic theory to break down if η is too large. Fig 8A and 8B demonstrate that the time course of the chain score in the stochastic simulations matches the deterministic theory very well when η is set such that convergence to synfire chains occurs over a time scale of ∼10 or ∼20 hours, respectively. Correspondingly, the time course of the chain score, when plotted as a function of ηt is nearly identical in these two conditions (red and green traces in Fig 8C), in very good agreement with the prediction of the deterministic theory (black trace). When the learning rate is faster, such that convergence to a synfire chain occurs over a time scale of ∼5 hours, the stochastic simulations are somewhat slower to converge than the deterministic simulation (blue trace in Fig 8C). At an even higher learning rate, in which perfect synfire chains emerge under the deterministic dynamics within ∼2 hours, the stochastic simulations do not converge at all to synfire chain structures within a comparable time frame (yellow trace).

In summary, Fig 8 demonstrates that STDP in a stochastic spiking network can lead robustly to synfire chain structures over a time scale of several hours. Moreover, if the learning rate is sufficiently small, such that the time scale of convergence is of order ∼10 hours or more, the time course of convergence is predicted very well by the deterministic theory of Eq 2. As another demonstration for the relevance of the deterministic theory, we measure in Fig 8D the dependence of the chain score on the synaptic latency in stochastic spiking simulations lasting 60 hours of biological time, using the same parameters as in Fig 7C and a learning rate that matches the blue trace in Fig 8C. The results are very similar to those obtained in the deterministic dynamics (Fig 7C).

Self organization into self connected assemblies

Finally, we check whether self connected assemblies can spontaneously emerge under the full STDP dynamics. The motif {1, 1} promotes formation of such structures (Fig 4B and 4D). Therefore, we choose biophysical parameters that increase the relative contribution of this motif. First, we choose an STDP function with a Mexican hat structure (Fig 9A), which increases synaptic efficacies between neurons that spike at similar times, regardless of the temporal order of the spikes. Second, we note that the contribution of the {1, 1} motif is independent of synaptic latency, because both the pre and post synaptic neurons i, j accrue the same latency relative to the source neuron k. On the other hand, coefficients of other low order motifs do depend on the synaptic latency (Fig 9B).

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Fig 9. Self organization into self connected assemblies.

A. STDP function with Mexican hat structure. B. The coefficients f_{α, β} of four different motifs as a function of the synaptic latency. Orange line—{1, 1}, black line—{2, 1}, gray line—{2, 0}, red line—{1, 0}. C. Emergence of self connected assemblies under stochastic dynamics. Example of connectivity obtained in a parameter regime in which self connected assemblies emerge robustly. Left—initial connectivity, right—steady state. Here the synaptic latency is 5.25 ms (for other parameters, see Methods).

https://doi.org/10.1371/journal.pcbi.1005056.g009

Based on the above reasoning, we expect that self connected assemblies will emerge, under the influence of STDP and synaptic competition, for finite synaptic latencies, in which contributions from motifs other than {1, 1} are suppressed. Fig 9C shows results from a stochastic simulation of a Poisson network, starting from initial random connectivity, with a synaptic delay of ∼5 ms. Here, a precise structure of self connected assemblies emerges robustly. Other structures were observed for alternative choices of the synaptic latency.

High order motifs beyond the third order

We focused on the influence of biophysical parameters on the contribution of motifs up to third order. However, Eqs 2 and 3 include also contributions from higher order motifs. This raises a question, why an analysis up to third order allowed us to predict the emergence of global structures. A partial answer to this question is that for small synaptic weights, the contribution of high order motifs decays with the number of participating synapses. One important factor contributing to the decay of motif coefficients is that when |α − β| is large the spike correlation function is shifted outside the range of the STDP window. In addition, spike correlation functions widen with increase of the number of participating synapses, thus decreasing their overlap with the STDP function.

Another, more formal argument is based on the derivation of Eq 3: as long as the synaptic weights are sufficiently weak, such that all eigenvalues of the connectivity matrix are smaller than unity, the expansion in Eq 14 converges for all ω, and therefore the sum in Eq 3 must converge as well (this is also the condition for stability of the linear neural dynamics). This implies that the combined contribution from all motifs of order n must decay as a function of n.

Even though contributions of high order motifs must eventually decay, our simulations of the full STDP dynamics were performed in a regime where motifs beyond the third order do influence the plasticity. To a large extent, the effect of higher order motifs can be predicted by the contribution of second and third order motifs (S5 Fig). For example, all high order motifs that satisfy α − β = 1 are expected to assist the formation of synfire chains structures, based on the same intuition that was demonstrated in Fig 4A. All these motifs also share a similar time course since they involve a delay of one synapse between the activity of the pre and post synaptic neurons. Similarly, all the motifs with α = β are expected to contribute to formation of self connected assemblies. In similarity to the motif {1, 1}, and in contrast to the other motifs, their contribution is not influenced by the synaptic latency. The similar dependence on the synaptic latency in motifs with the same value of α − β is illustrated in S5 Fig.

Formation of synfire chains in previous works

Fiete et al. [25] demonstrated that narrow synfire chains, in which single units project sequentially to each other, can emerge spontaneously under the combined influence of STDP and heterosynaptic competition. However, wide synfire chains did not emerge unless correlated inputs were fed into the network, even though the STDP simulations implicitly included motifs of all orders. Our results suggest why it was difficult to obtain wide synfire chains robustly in this work: first, Fiete et al. did not include in their model self depression, which can suppress the contribution of the first order motif, bringing the plasticity dynamics to an appropriate regime (see Fig 6A). In particular, it is likely that the choice of parameters was such, that the relative contribution of the third order motif was not sufficiently large.

Interestingly, wide (but sparsely connected) synfire chains were spontaneously produced in another recent work [44], which considered a simplified model of neural and synaptic dynamics, operating in discrete time bins. By applying our framework to this model, it is straightforward to see that only a small subset of the possible structural motifs contributed to plasticity, due to the simplified and discrete dynamics. In addition, the synaptic plasticity rules included an effective form of self-depression. Thus, the spontaneous formation of synfire chains in [44] is consistent with the predictions of our work.

Distribution of synaptic weights

Considerable theoretical attention has been devoted to the influence of STDP on the steady state distribution of synaptic weights [12, 13, 45–49]. This interest is partially motivated by the observation in specific brain areas of unimodal distributions of synaptic efficacies, often following approximately a log-normal distribution [1, 50]. Due to our interest in formation of synfire chains and self-connected assemblies, we focused on situations in which the steady-state weight distribution is bimodal. However, under certain choices of parameters which do not lead to the formation of ordered structures, we observe unimodal weight distributions (see, for example, the black area in Fig 6A which is characterized by strong synaptic self-inhibition [48]). It will be of interest in future studies, to ask whether it is possible to obtain highly ordered structures, in which the non-vanishing weights follow broad distributions, perhaps under a softer implementation of the synaptic competition.

It will also be interesting in future studies to consider situations in which connections exist between a subset of neuron pairs: for example, the structural connectivity may be sparse. The analytical framework that we developed can be directly applied to networks with an arbitrary adjacency matrix. In this case, only efficacies of structurally existing synapses should be updated based on Eqs 2 and 3 (note that in Eq 3, only those motifs that are realized in the structural connectivity graph can contribute to the sum, since the synaptic efficacies associated with non-existing connections vanish). Moreover, the formalism can be easily generalized to consider synapses with heterogeneous biophysical properties.

Significance for neural dynamics and structure

Nucleus HVC plays a key role in timing the vocal output of songbirds [21, 22]. This nucleus is a compelling candidate for a brain area that can organize autonomously to produce structured dynamics, since auditory deprived songbirds generate a song with a stereotypical temporal course [51, 52]. However, in almost all theoretical works that addressed how local plasticity rules give rise to temporal sequences of neural activity, it was necessary to provide some form of structured input into the network in order to robustly produce the sequential neural activity [25, 53–55]. Similarly, structured inputs were required in order to robustly produce self connected assemblies, which give rise to another useful form of neural dynamics, characterized by multiple stable states [27–31]. It is therefore significant that synaptic structures which support structured neural dynamics can emerge in a neural circuit without any preexisting order in the synaptic organization, and without any exposure to external stimuli. Appropriate choices of the biophysical parameters, which enable this type of autonomous organization, became apparent by applying the theoretical formalism and reasoning developed in this work.

Finally, we briefly mention another area of future work, in which the formalism developed here may find a useful application: we expect that the analysis of synaptic dynamics in terms of contributions from structural motifs, will be valuable for assessing the role of STDP in shaping the high order statistics of cortical connectivity, as experimental data on these statistics become increasingly available [1, 2, 32].

Network dynamics

The time dependent activity of neuron i is a stochastic realization of an inhomogeneous Poisson process, with expectation value (5) where N is the number of neurons, W is the connectivity matrix, a(t) is the synaptic current, b_i is a constant external input, and is the spike train of the neuron k (where are spike times of the neuron). We assume that the neurons do not excite themselves, meaning that ∀i W_ii = 0.

An analytical expression for the synaptic drift

Assuming that all spike pairs contribute to STDP, the change in the synaptic efficacies due to STDP can be expressed as follows: (6) where (7) is the change in synaptic efficacy arising from spikes in the post synaptic neuron i at time t, and (8) is the change following a spike in the pre synaptic neuron j at time t. In both terms, the integration is over all previous spikes of the presynaptic neuron (Eq 7) or the postsynaptic neuron (Eq 8).

We define the correlation function of spikes in each pair of cells as follows, (9) where 〈 ⋅ 〉 denotes averaging over different realizations of the Poisson dynamics for a given connectivity. For constant external input, and under the assumption of slow learning, the correlation function is stationary, and does not depend on t. We denote the rate of change in the synaptic efficacy, averaged over realizations of the Poisson dynamics as (10) and refer to it in short as the synaptic drift. Using the correlation function we can express the synaptic drift as follows, (11)

For linear dynamics the correlation function can be written exactly. In the frequency domain [36], (12) where r_i = 〈λ_i〉, the diagonal matrix D_ij = δ_ij r_j, I denotes the unit matrix, and we use the convention in which the Fourier transform of a function g(t) is defined as .

The average firing rate can be easily obtained from Eq 5: (13) By substituting Eq 12 in Eq 11, we obtain Eq 2. Note that the diagonal terms should be ignored, since we assume that there is no synapse from a neuron to itself.

Power series expansion of the average learning dynamics.

The matrix inverses appearing in Eq 12 can be expanded using the matrix identity [37] (14) In our context, this substitution can be seen as an expansion in powers of the synaptic efficacies. We assume that the synaptic efficacies are sufficiently weak, such that for all values of ω, all eigenvalues of are smaller (in absolute magnitude) than unity. Note that below we normalize the synaptic current such that , and for all ω. In this case the requirement is that all eigenvalues of W are smaller (in absolute magnitude) than unity.

Using Eq 14 we rewrite Eq 2 as Eq 3, where we define: (15) and (16) Note that f_{α, β} are dimensionless, and f₀ has dimensions of time. In the time domain, (17) where c_{α, β} can be written as a series convolutions of the synaptic current function: (18) The way f_{α, β} is determined from the synaptic current and the STDP function is illustrated in Fig 10 for a few examples.

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Fig 10. Dependence of motif coefficients on the time course of STDP and time course of post synaptic currents.

A-F. Examples illustrating how the coefficients f_{α, β} are determined for three motifs. Solid lines: motif driven spike correlations (Eq 18). Dashed lines: the STDP function (chosen to be antisymmetric). Yellow area: the integral of the product of these two functions, which determines the motif coefficient. For simplicity, we consider only the correlation structure induced by the excitatory synaptic connectivity. A. The correlation time course (solid line) induced by the motif {1, 0} is simply the synaptic current function. B. In comparison with A, the correlation time course induced by the motif {2, 0} peaks later and is more widespread, because the coupling between the pre and the post synaptic neurons is mediated through an intermediate neuron. C. The correlation function induced by the third order motif {2, 1} peaks at a similar time as in A, because in both motifs α − β = 1, meaning that the paths leading to neurons i and j differ in length by one synapse. D-F. A synaptic latency of 3 ms affects the overlap between the correlation time course and the STDP function, and thus the motif coefficients (see also Fig 5). D. In the motif {1, 0} the correlation time course is shifted by 3 ms relative to panel A, leading to a decrease in the overlap with the STDP function and a decrease in f_1,0. E. In the motif {2, 0} the spike correlation function (solid line) is shifted by 6 ms relative to panel B because two synaptic latencies accumulate along the path from the pre synaptic neuron j to the post synaptic neuron i. The overlap with the STDP function and the coefficient f_2,0 are sharply reduced. F. In the motif {2, 1} the spike correlation function is shifted by 3 ms relative to panel because the relative latency, incurred along the paths from the source neuron to neurons i and j amounts to a single synaptic latency. The time course of correlations arising from this motif (solid line) is very broad, and therefore shifting it relative to the STDP trace has relatively little influence on the overlap with the STDP function and on f_2,1.

https://doi.org/10.1371/journal.pcbi.1005056.g010

Full plasticity dynamics

We consider N excitatory neurons, with modifiable recurrent connections. In addition to the STDP, the excitatory synapses undergo heterosynaptic competition, and possibly constant self depression and growth at a constant rate. The full plasticity dynamics of these synapses can be summarized by the following expression: (19) The terms represent the heterosynaptic competition [25]: competition over the input to neuron i, (20) and competition over the outputs from neuron j: (21) Here, Θ(x) is the Heaviside step function: When ψ is sufficiently large, the competition guarantees that the sum over each row and column of W^ex does not exceed W_max. Finally, the term represents self depression (constant weakening of the synapse in proportion to the synaptic efficacy), and the term γ represents a constant growth of each weight at a fixed rate.

In addition to these rules, the excitatory synapses are restricted to the range [0, w_max].

Simulations

Power series expansion simulations.

In Figs 4, 6, 7B and 7C (Gray and red lines), and in S2 Fig, STDP is modeled as in Eq 3, but with a small number of non-vanishing coefficients f_{α, β} which are set as in Table 1. In Fig 7B and 7C the coefficients f_{α, β} which are not set to zero are directly evaluated from the STDP function and the synaptic current function.

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Table 1. The coefficients f_{α, β} in Figs 4, 6 and S2.

https://doi.org/10.1371/journal.pcbi.1005056.t001

Simulations parameters.

Synaptic current function: In all simulations, (24) where τ₁ = 5ms, τ₂ = 1s, and d is the synaptic latency. The coefficient a₀ normalizes the synaptic current such that . This choice sets a meaningful scale for W: if W_ij = 1, a single spike in a pre synaptic neuron j increases (or decreases), on average, the number of spikes emitted by the post synaptic neuron i by one.

STDP function: In Figs 5, 7, 8, 10, S1 and S3 we used the following antisymmetric STDP function: (25) where: (26) h₀ = 10⁴, τ₁ = 3ms, τ₁ = 2sec, and , A₋ = −A₊.

In Fig 9 we used a symmetric STDP function: (27) where σ = 12ms, A = 5.2 ⋅ 10⁴.

In S4 and S5 Figs we used the following asymmetric (but not antisymmetric) STDP function: (28) where: (29) , , , τ₂ = 2sec, , and .

Limitation on the total synaptic input and output: In all simulations, (30) The parameter M affects the number of neurons in each group when the ordered structures emerge [25].

Learning rate: In all simulations except the stochastic simulations, the learning dynamics were implemented using the Euler method with an adaptive time step, chosen such that the maximal change in the weights in each step was 0.0005 (Fig 4), 0.0001 (Figs 6 and S2) and 0.02 (Figs 7, S3 and S4), and 0.002 (Fig 8).

Initial connectivity: The initial weights were chosen independently from a uniform distribution between 0 and w_max M/N (Figs 4, 6, 9 and S2), and between 0 and 1.5w_max M/N (Figs 7, 8, S3 and S4).

Simulations duration and convergence criterion: The convergence criterion was that in the last 10 iterations the absolute change in each element of the matrix did not exceed e⁻¹⁵. The stochastic simulations were conducted using stochastic Euler dynamics with a time step of 0.25 ms.

Other parameters are summarized in Tables 2 and 3.

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Table 2. Simulation parameters.

https://doi.org/10.1371/journal.pcbi.1005056.t002

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Table 3. Supplementary parameters.

https://doi.org/10.1371/journal.pcbi.1005056.t003

Chain score and ordering of connectivity matrices

To classify groups of neurons that share similar connectivity, we performed k-means classification on a set of N vectors, where the i-th vector includes all the excitatory input and output synaptic efficacies of neuron i: , and using a squared Euclidean distance. We then reordered the neurons (and the connectivity matrix) based on the groups identified by the k-means clustering. When searching for synfire chain structure, we chose the order of groups as follows: We randomly chose one group and set it as the first group. We then looked for a remaining group that receives the largest total synaptic input from the first group, and set it as the next group. This process was repeated to include all groups. Next, we compared the ordered connectivity matrix to an “ideal” binary connectivity matrix that represents complete feed forward connectivity between the groups, or complete clustering into self connected groups. The chain score is defined as: 1 − x, where x is the normalized square distance between the ordered matrix, scaled by the largest element, and a matrix representing ideal feed forward connectivity (or a perfect arrangement of self connected clusters). Finally we maximized the similarity score over a range of values of k.