The evolution of activity variables/firing rates

As in [70, 71] with somewhat different notations, the system describing the dynamics of N neurons (populations) is as follows (1) where u_j is the activity variable (average membrane potential) of neuron j, I is a constant external input, x_j is the firing rate of neuron j, the coefficients J_ij express the strength of the excitatory connections from neuron j to neuron i and τ is the time constant, measured in miliseconds. The terms represent inhibition, discussed in more detail below. The firing rate is itself a monotonously increasing function of the activity variable with limiting values 0 and 1. This function is often taken as x = g(u) = (1 + e^−u/μ)⁻¹. In this work we will use an approximation of g, as shown below.

System (1) can be expressed in terms of firing rates by means of the transformation x_i = g(u_i). Learned patterns are steady state patterns of Eq (1) of the form (ξ₁, …, ξ_n), ξ_j = 0 or 1. In order to apply the linearized stability principle we need to be able to evaluate partial derivatives of the right hand side of Eq (1) at the learned patterns. However for such states the derivatives do not exist for the particular choice of g. This makes it impossible to apply the algebraic method of linearization (computation of eigenvalues of the linearized system) in the current models. We can remedy this using the approach introduced in [91] by replacing the function g⁻¹(x) = ln x − ln(1 − x) by its Taylor expansion f_q(x) at x = 1/2 up to some arbitrary order q. When we let q tend to infinity f_q tends uniformly to g⁻¹ in any interval in (0, 1). In the following, for simplicity, we take the expansion to first order f₁(x) = 4x − 2, (this corresponds to q = 1). A different choice of q would not significantly alter our results. After renaming the parameters we arrive at the equations (2) The system (2) has the following fundamental property: any vertex, edge, face or hyperface of the cube [0, 1]^N is flow-invariant: trajectories with starting point in any one of these sets are entirely including in it. This implies in particular that the vertices are equilibria, or steady-states, of Eq (2). The vertices have coordinates 0 (inactive unit) or 1 (active unit). Hence vertices correspond to patterns for the neural network and whenever a vertex is a stable equilibrium it represents a learned pattern.

The modelling of synaptic depression

According to the synaptic depression assumption the coefficients J_ij vary in time according to the rule: (3) where the evolution of the synaptic variable s_i is given as follows [102] (4) τ_r and U being the time constant of the recovery of the synapse and the maximal fraction of used synaptic resources.

The modelling of excitatory connections

We assume that the matrix of excitatory connections (J^max)_{i, j = 1, …n} is derived from a set of learned patterns which must be stable steady states of the system. Following [71] we use the Hebbian learning rule for sparse matrices, as introduced in [103], see also [104]. According to this rule the coefficients of the connectivity matrix (J^max)_{i, j = 1, …n} (without synaptic depression) satisfy (5) where ξ¹, …, ξ^P are the learned patterns, N is the total number of neurons and p is the ratio of active to inactive units, measuring the sparsity of the matrix J. Note that the matrix given by Eq (5) is symmetric.

Let us set ν = (Np(1 − p))⁻¹. We simplify the expression (5) by introducing a change of variables and parameters, see also [104]. The rhs of Eq (2) can be rewritten as where now (6) Remark that J^max is symmetric, while J at t > 0 will not be so (as long as some and not all neurons are active, see Eq (3)). In other words synaptic depression has the effect of breaking the symmetry of the connectivity matrix.

We rename parameters μ/ν as μ, etc., rescaling time by t = t′/ν and update the definition of J_ij in Eq (2) by , with given by Eq (6). Further, we assume that the connectivity matrix is sparse, which is consistent with neurophysiological data [105], as well as with computational models showing that a sparse matrix allows maximal storage capacity [106–110]. In the following we shall assume that p ≪ 1 (sparse matrix) and replace p by 0 in Eq (6). This guarantees that the weights are positive, which is consistent with the assumption that they correspond to the excitatory connections. Moreover, given that ν is a constant between 0 and 1 and, due to the sparsity of , is not particularly close to 0, we set, for simplicity, ν = τ. This choice does not qualitatively alter our results. Hence the context of our study is system (2) with τ = 1 and the goal is to find latching dynamics between learned patterns with the connectivity matrix given by Eq (6). As an intermediate stage of our investigation we will consider systems of the form (2) with weights that do not satisfy Eq (6).

The modelling of inhibition

The term −I in Eq (2) corresponds to constant (tonic inhibition). Due to the presence of this term the pattern consisting of all neurons inactive is stable.

The term − λ∑x_i is the non-selective inhibition, depending on the activity of the specific neurons. This contribution should be thought of as feedback inhibition: a pyramidal neuron which is active excites some interneurons which contribute an inhibitory feedback. The choice of the dense inhibitory connectivity is supported by the experimentally known fact that interneurons are characterised by an extensive axonal arborisation, which allows each one of them to reach a large number of pyramidal cells in a local network [111].

The modelling of noise

Noise plays a very important role of facilitating the transition from steady states that lose stability to the ones that follow in the sequence of latching dynamics. Therefore the noisy perturbation should not have the factor of x(1 − x), which would make it very small near the vertices, yet it must preserve the invariance of the cube [0, 1]^N. We construct the noise term starting with white noise and subsequently modify it so that it points towards the interior of the cube. This noise term can be thought of as a fluctuation of the firing rate due to random presence or suppression of spikes. The role of the adjustment brought to white noise is to ensure that negative firing rates or firing rates greater than 1 do not arise. In practice, in our simulations we add a noisy perturbation to the initial condition at regular intervals of time, making sure that the perturbations are positive for firing rates near 0 and negative for firing rates near 1.

Sparse connectivity and the non-selective inhibition imt stable patterns contain only a few active neurons.

Heteroclinic chains

One of the goals of this work is to show that latching dynamics is approximated by heteroclinic chains, which we introduce here. Given given a sequence of steady state patterns ξ¹, …, ξ^M, M < P, a heteroclinic chain consists of a sequence of connecting trajectories (dynamic transition patterns) between these patterns and of a sequence of time instances 0 < t₁ < ⋯ < t_M such that the transition from ξ_k to ξ_k+1 exists for the coefficients of the connectivity matrix J_ij evaluated at t_k, that is , where i, j = 1, 2, …, n.

We will show that latching dynamics is closely approximated by heteroclinic chains. Hence, finding a heteroclinic chain in our model becomes equivalent to the problem of finding heteroclini chains. This problem can therefore be formulated as follows:

Problem: let there be given a sequence of patterns ξ¹, …, ξ^M, M < P, where ξ^k and ξ^k+1 share at least one active unit for all k = 1, …, P − 1. Under which conditions does there exist a sequence of connecting trajectories ξ¹ → ⋯ → ξ^P, so that a heteroclinic chain is realized between these patterns?

Eigenvalue computations

The structure of Eq (2) makes the eigenvalues of the system linearized at each steady state pattern lying on a vertex of the hypercube [0, 1]^N easy to compute (diagonal Jacobian matrix). Let ξ = (ξ₁, …, ξ_N) be a vertex (hence ξ_j = 0 or 1), then the eigenvalue at ξ along the coordinate axis x_k has the form (7) The stability condition is now

(S) σ_k < 0 for all k = 1, 2, …, n.

Note that this algebraic method would not be available if we had not replaced the function f, equal to the inverse of the transfer function, by its Taylor polynomial.

The assumption of sparsity implies that for each k only a few ’s can be non-zero. This means that in a stable pattern only a few ξ_j’s can be non-zero, otherwise the contribution of the non-selective inhibition would not allow the stability condition to hold.

Formula (7) and the stability condition (S) are the tools that will allow us to create conditions for the existence of heteroclinic chains and establish the role of the overlap between learned patterns. We show that such overlap is needed for the existence of a heteroclinic chain.

Sparse coding and the relation between latching dynamics and heteroclinic chains

Consider a pattern ξ = (ξ₁, ξ₂, …, ξ_N) and suppose ξ is a stable steady state of Eq (2) with s_j = 1, j = 1, …, N. We will argue that only a few of the components ξ_j can be significantly larger than 0. Assume this is not the case. Then, either there is a large number of ξ_j’s satisfying 0 < ξ_j < 1 or there is a large number of ξ_j’s equal to 1. In the first case λ∑ξ_j must be larger that ∑J_ij ξ_j, so that the pattern cannot be a steady state. If the second case arises then the pattern can be a steady state but it cannot be stable, since the eigenvalues corresponding to the entries equal to 1 must be positive, see Eq (7). Hence the number of components that are not very close to 0 must be small. This property is an expression of sparse coding (few neurons code for every concept), which, in the context of our model, is a consequence of sparse connectivity in the excitatory network and dense connectivity of feedback inhibition.

The above argument implies that latching dynamics must necessarily take place near the surface of [0, 1]^N, as the learned patterns have to be close to the surface and the transitions between them cannot be take a very long time, otherwise synaptic depression would weaken the excitatory connections making it unlikely for such transitions to occur. Hence the question is to identify dynamics on the boundary of the cube that could be a good approximation of latching dynamics. In fact the argument of sparse coding given above suggests that the dynamics is restricted to a subset of the boundary of the cube of relatively small dimension. In a heteroclinic chain we find a model of the dynamics in the edges of the cube which, when perturbed by noise gives a faithful representation of latching dynamics.

Constructing heteroclinic chains

Due to the action of synaptic depression each of the learned patterns in a heteroclinic chain must lose stability due to one or more of the eigenvalues σ_k becoming positive. We assume that no two eigenvalues become positive at the same time, which implies that a noisy trajectory must follow the direction of the unstable eigenvalue. We will assume that in order to pass from one learned pattern to the next, the trajectory follows the edge corresponding to the unstable eigenvalue to the opposite vertex, which is a saddle point with a single unstable direction connecting to the next learned pattern in the chain. The chain will therefore be a sequence of elementary chains consisting of connections with three elements: a learned pattern ξⁱ that becomes unstable due to synaptic depression (prime), a transition pattern of saddle type, and the next learned pattern ξⁱ⁺¹ (target). The active units in the pattern correspond to the overlap between the patterns ξⁱ and ξⁱ⁺¹ The fact that the transition pattern should be unstable imposes another condition on the eigenvalues. These conditions will be presented in the next section.

We argue that an elementary chain is the most likely mechanism of transition from ξⁱ to ξⁱ⁺¹. It is certainly the simplest case dynamically. Any more complicated dynamics would be likely to increase the passage time, so that the target pattern could lose stability due to synaptic depression before becoming active in the chain. Finally, more complicated dynamics would require the existence of additional unstable eigenvalues leading to additional constraints on the matrix .

Constraints on the connectivity matrix

We now state the algebraic constraints from the eigenvalue conditions (7) which define the parameter regions where heteroclinic chains could exist (see S1 File for a derivation of these conditions). These conditions are only necessary, in fact our numerics show that heteroclinic chains which follow a prescribed sequence of connections arise in a reliable manner in yet smaller parameter regions. The origin of these conditions is the requirement of stability of the steady states in the absence of synaptic depression combined with the requirement of the existence of a passage to the next steady state once the state currently attracting the dynamics loses stability due to synaptic depression. A cycle we consider joins a sequence of learned patterns ξ¹ → ⋯ → ξ^p such that each of them has exactly m excited neurons (with entry 1) and the switching from one pattern to the next corresponds to switching the values in two entries. Possibly after re-arrangement of the indices it is no loss of generality to assume that In addition we have p − 1 transition patterns We make a simplifying assumption that the entries of J^max are 0 outside of a band around the diagonal of width 2m − 1 (this is consistent with the requirement of the sparsity of the matrix). We introduce: (8) The requirement that the patterns ξ¹, …ξ^p are stable in the absence of synaptic depression can be expressed, using Eq (7), by the condition (9) (10) Other types of constraints come from the fact that, in the time interval of transition from one pattern to the next, the dynamics must approach a transition state from the direction of the prime pattern and leave in the direction of the target pattern. It means that there is a time instance such that (11) which implies a weaker condition (12) The combination of Eqs (9), (10) and (12) places severe restrictions on the parameters λ, I and . Additional constraints can be derived from the fact all the other directions of have to be stable, in order to ensure the reliability of the cycle, but we did not explore these conditions here. For m = 2 we can use Eq (11) and the fact that there is only one synaptic variable pertaining to to obtain the inequality: . From the symmetry of the connectivity matrix we conclude that . In other words, the elements on the upper diagonal and the lower diagonal must be increasing. This, combined with Eqs (9), (10) and (12), gives, for m = 2 (13) The property of increasing diagonal elements, in practice, prevents the existence of long chains as the large coefficients will activate the corresponding neurons just due to the presence of noise.

We use Eq (13) to select the parameters for our numerical examples. For the details of the derivation of the algebraic constraints we refer to S1 File.

Conditions based on slow/fast dynamics

To take advantage of the fact that the synaptic variables s_i are slow compared to the firing rates we write the equation for the s_i’s in the form (14) with In this formulation the model has a time scale separation and ρ is a regular parameter.

The transitions are governed by Eq (2), which implies that as ε → 0 each one of them lasts for an approximately constant positive amount of time. It follows that as ε → 0 the change in s_i in the transition period tends to 0, that is s_i remains close to a constant value, approximated by bifurcation points, corresponding to the loss of stability of ξ_i. These values can be derived in a recurrent manner as follows:

1. At t = 0 s₁ = s₂ = 1. Until the loss of stability of ξ₁ it holds that x₁ ≈ 1 and x₂ ≈ 1. Hence, in that period, we set: x₁ = x₂ = 1 and Eq (14) becomes (15) where i = 1, 2. Since s₁ and s₂ have the same initial condition they remain equal. Using Eq (7) we derive that time when ξ₁ loses stability is defined by: Hence the bifurcation is given by the s₁ and s₂ values:
2. Assuming that s₂ does not change during the transition , at the beginning of the period when the trajectory is near ξ₂, we have and s₃ = 1. The evolution of (s₂, s₃) until the next stability loss is Eq (15) with i = 2, 3. Note that Eq (15) is a linear equation and hence can be solved: (16) By Eq (7) we must now choose t_B so that Hence the values of (s₂, s₃) corresponding to the point of the loss of stability of ξ² are and . To obtain and we solve the linear system (17) with . The values of and can now be found by solving this linear equation (the determinant of the matrix is easily shown to be non-zero).
3. The process of the derivation of the bifurcation values is iterative. If is known then and are obtained by solving the linear equation (18) This calculation allows us to check for a given matrix J, if the specified heteroclinic chain exists for sufficiently small ε. The first set of conditions is (19) The second set of conditions is obtained based on the requirement that, for each i = 1, 2, …, P, at the ith bifurcation point the following properties hold:
   • ξⁱ⁺¹ must be stable,
   • must be a saddle with the direction of e_i stable (e_i is the vector with ith component 1 and the other components 0),
   • ξⁱ must be stable in the direction of e_i+1.

Explicit conditions can be derived in an iterative manner using Eq (7), in the sequel we do this numerically in the context of specific examples. If these conditions are satisfied then a chain exists for sufficiently small ε and small noise. If the conditions are violated then the chain does not exist unless the noise is large and the transitions are driven exclusively by noise.

Satisfying the learning rule

Our results show that, given a small network, the parameters have to be tuned quite precisely to obtain a heteroclinic chain. Adding the requirement that the matrix is obtained using the learning rule (6) gives an even more severe constraint. We have constructed examples of networks supporting heteroclinic chains with each neuron involved in some of the patterns forming the chain. In each case the connectivity coefficients we used had larger values than given by the patterns involved in the chain alone. To solve this problem we designed a method of defining a larger system, with the connectivity matrix of the form (20) and added learned patterns which do not participate in the chain but with an overlap with the patterns forming the chain, so that the matrix is obtained using the learning rule (6). The matrix A consists of many blocks with few non-zero coefficient that are small in comparison to the entries of . The matrix B is block diagonal, with the off-diagonal entries in each of the blocks equal to 1. The added learned patterns must satisfy the constraints (9) and (10) to ensure their stability. This way the matrix is sparse (about 25% non-zero elements in the example we constructed). There is no natural algorithm to construct , so we refrain from making any further specifications. We constructed for a specific example, see S2 File.

A simple example—An elementary chain

In this section we examine the simplest case of a network of three neurons and two learned patterns (21) This example is the prototype of an elementary chain. In this section we describe the simplest case of constructing a heteroclinic chain, namely a sequence of connecting orbits (dynamic transition patterns) along edges of the cube in connect these patterns in a chain. This requires the presence of one intermediate equilibrium , which by assumption is not a learned pattern and has an unstable direction along the coordinate which passes from 0 to 1 in the sequence (22) Computing the connectivity matrix from the learning rule (6) with patterns ξ¹ and ξ² is straightforward and gives (23) Applying formula (7) with N = 3 and J^max given by Eq (23) it is easily checked that the two learned patterns have negative eigenvalues, hence are stable, in the absence of synaptic depression, iff 1 < I + 2λ < 2 − μ. This is our first requirement.

In the remaining of this section , resp. , will denote the eigenvalue at ξ_i, resp. along x_k.

We now “switch on” synaptic depression. As time elapses, synaptic weights will be modified according to Eq (4). For a given pattern (steady-state of Eq (2)) on a vertex of the cube [0, 1]³) the evolution of the synaptic variable s_i (i = 1, 2 or 3) can be of two types: (i) if x_i = 0 the value s_i = 1 is a steady-state of Eq (4); (ii) if x_i = 1 then s_i decreases monotonically towards the limit value S = (1 + τU)⁻¹ which is the steady-state of Eq (4) at x_i = 1.

Let us describe in terms of the eigenvalues associated with each pattern, the scenario which would lead to the expected heteroclinic chain.

First, the weakening of the synaptic variable s₁ should modify the stability of the learned pattern ξ₁ in such a way that a trajectory will appear connecting ξ₁ to the intermediate along the first coordinate x₁. This entails that the eigenvalue at ξ₁ (eigenvalue along coordinate x₁), which initially is negative, becomes positive after some time. This is a kind of dynamic bifurcation, in which time plays the role of a bifurcation parameter through the evolution of the synaptic variables. Simultaneously we want the eigenvalue at become negative at finite time so that this state is attracting along direction x₁. By Eq (7) a sufficient condition for this is .

The second step is to check conditions which would allow for the existence of a trajectory connecting to ξ₂ along the edge with coordinate x₃. This requires that the eigenvalue be positive after some time (possibly already at t = 0), hence by Eq (7), . The two conditions we just derived are incompatible with matrix Eq (23), so we can conclude that this matrix does not admit the heteroclinic chain Eq (22).

From the above discussion we can infer that a necessary condition for the existence of a chain Eq (22) is that . Since J^max is symmetric this means that its upper diagonal must have strictly increasing coefficients. In addition to these conditions we also request that: (i) ξ₁ be “more” stable in the x₂ direction than in the x₁ direction, i.e. at t = 0, so that a trajectory starting close to this equilibrium will first destabilize in the x₁ direction (it may of course not destabilize at all); (ii) the x₂ direction at is stable; (iii) ξ₂ is stable when t large enough. This imposes additional conditions on the coefficients of J^max. Let us show that the matrix (24) satisfies all these conditions and hence admits the heteroclinic chain Eq (22). Of course this is an ad’hoc construction, however we shall show in S2 File that Eq (24) is a submatrix of the connectivity matrix of a subnetwork of a large sparse network under the learning rule (6) (see Eq (20)).

For Eq (24) the eigenvalues computed using Eq (7) are as follows: From this we obtain the following conditions:

1. 2 < I + 2λ < 3 (stability of the learned patterns in absence of synaptic depression),
2. 3S < I + 2λ ( becomes >0 in finite time),
3. 1 < I + λ (),
4. I + λ < 2S ( becomes >0 in finite time),
5. I + 2λ < 2S + 2 ().

These conditions are all satisfied if τ and U are chosen such that 2/3 < S and the point (λ, I) lies in the triangle bounded by the lines I > 0, 2 < I + 2λ and I + λ < 2S. The value of S = (1 + τU)⁻¹ being given, these conditions can be conveniently represented graphically. Fig 1 shows an example with τ = 100 and U = 0.004.

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Fig 1. The domain of values (λ, I) which allow for existence of a heteroclinic dynamics with matrix Eq (24) and S = 1/1.4.

https://doi.org/10.1371/journal.pone.0183710.g001

Let us illustrate numerically this result. We have integrated the Eqs (2) and (4) with N = 3 and coefficient values τ = 100, U = 0.004, I = 0.15 and λ = 1.2. Fig 2 shows a time series of x_I(t) (upper figure) and s_i(t) (lower figure) with an initial condition starting close to ξ₁. In order to observe the transitions Eq (22) in reasonable time we have incorporated a noise in the code, in the form of a small random deviation from initial condition on the x_i variables at each new integration time (simulation with Matlab).

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Fig 2.

Panel (a): time series of x₁ (blue), x₂ (red) and x₃ (green). Panel (b): time series of s₁ (blue), s₂ (red) and s₃ (green).

https://doi.org/10.1371/journal.pone.0183710.g002

A case with five neurons and the extended network of 61 neurons

We consider (25) with I = 0.3, λ = 3.4, μ = 3.1, τ_r = 400 and U = 0.01. This matrix and these parameter values meet all conditions for the existence of a heteroclinic chain joining the patterns ξ¹ = (1, 1, 0, 0, 0), …, ξ⁴ = (0, 0, 0, 1, 1), however Eq (25) does is not derived from the learning rule (6). In Fig 3 we show a simulation of a chain of four states existing for the above parameters.

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Fig 3. The chain of four patterns in the network of 5 neurons with the transition matrix Eq (25) and the parameters I = 0.3, λ = 3.4, μ = 3.1, τ_r = 400 and U = 0.01.

Panel (a) shows the evolution of the firing rate variables, panel (b) shows the evolutions of the synaptic variables.

https://doi.org/10.1371/journal.pone.0183710.g003

We extended this network to a network of 61 neurons with the connectivity matrix satisfying the learning rule (6), using the approach described earlier. The details of the construction can be found in S2 File. Fig 4 shows a simulation for the required chain joining the states whose first five components are ass ξ¹ = (1, 1, 0, 0, 0), …, ξ⁴ = (0, 0, 0, 1, 1) specified above and the remaining 56 components are 0.

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Fig 4. The chain of four patterns in the network of 61 neurons obtained by extending the five neuron example with the transition matrix Eq (25).

The parameter settings are as the same as in the simulation of Fig 3. Panel (a) shows the evolution of the firing rate variables, panel (b) shows the evolutions of the synaptic variables.

https://doi.org/10.1371/journal.pone.0183710.g004

Reliability of the chain.

We have computed the reliability of the chain with the prescribed sequence (1, 1, 0, 0, 0) → (0, 1, 1, 0, 0) → (0, 0, 1, 1, 0) → (0, 0, 0, 1, 1) in the extended network of 61 neurons, with respect to the parameter U (maximal fraction of used synaptic resources). For parameter values distributed in the interval 0.0005 ≤ U ≤ 0.110 we performed 10 simulations with the same initial conditions for each of the chosen parameter values. Fig 5 shows the distribution, for different values of U, of the proportion of the simulations for which a given pattern was activated in the chain.

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Fig 5.

Panel A. Reliability of the chain of four patterns in the network of 61 neurons as a function of U. Panels B1-4. Activities of neurons coding for the patterns in the chain as a function of time for four representative values of U.

https://doi.org/10.1371/journal.pone.0183710.g005

Panel A of Fig 5 shows that the activation of the full length chain (all of the four successive patterns) is possible for a limited range of values of U (from .0081 to .0093, see panel B3). Even within this range of values of U and with fixed parameter values, the chain does not always fully develop on every simulation due to the presence of noise. Further, for values of U lower or higher than this range, the chain does not fully develop for two different reasons. On the one hand, when U decreases, the length of the activated chain decreases because the network stays in the state corresponding to the first pattern (panel B1) or to the second pattern with slow transition time (panel B2). On the other hand, when U increases, the length of the activated chain decreases because the network does not stay in the state corresponding to the first but does not activate the second pattern either, ending in a state where no neuron is activated (panel B4). Taken as a whole, these results show that the value of U determines the reliability of the chain in terms of number of patterns activated, with the patterns occurring later in the chain less likely to be activated. Further, the value of U also determines the state of the network after the first pattern, ranging from this first pattern or the different following patterns (low values of U) to an absence of activity of the neurons (high values of U). For the optimal range of values of U, the reliability of the chain is maximal but not perfect due to the presence of noise.

Slower synaptic depression leads to more reliable chains.

Here we show some simulation complementing our analysis of the effect of varying the speed of the evolution of the synaptic variables s_j. Earlier we introduced new parameters and ρ = τ_rU and studied the effect of decreasing ε, while keeping ρ fixed, which is equivalent to simultaneously increasing τ_r and decreasing U. We showed that this has the effect of decreasing the threshold of the noise amplitude needed for activation of a heteroclinic chain. This result is confirmed by simulations, as shown in Fig 6. In addition, our simulations show that the curve marking the upper limit of the noise window for which heteroclinic chains are activated also increases with the decrease of ε. Altogether, the window of noise for which the chains are activated is larger if the synaptic variable evolves more slowly. Our hypothesis, based on the simulations shown in Fig 6, is that time scale separation increases the reliability of the deterministic fast component of the dynamics, limiting the role of the stochastic component to the initiation of the transitions from one state to the next.

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Fig 6. Activation of heteroclinic chains as a function of the time scale separation ε = 1/τ_r and the amplitude of the noise σ.

The top panel shows the simulations marked by dots in the (ε, σ) plane, colour coded according to the extent and reliability of the activation of the heteroclinic chain, as indicated by the colour bar on the right. Lower panels show sample time traces corresponding to a selection of the parameter points, as indicated.

https://doi.org/10.1371/journal.pone.0183710.g006

A chain connecting six patterns with m = 2

In this section we present an example of a longer chain involving six neurons and 5 patterns. We do not construct the extension to a large network with a matrix derived from the learning rule (6). This would be possible using the approach outlined earlier and carried out in detail for the example of five neurons in S2 File. However the matrix we consider has larger entries than the one of the preceding section (five neurons), which implies that a larger extended network would be needed.

We consider the following connectivity matrix: (26) This matrix satisfies the necessary conditions eq-condsmt based on the learned patterns ξ¹ = (1, 1, 0, 0, 0, 0), ξ² = (0, 1, 1, 0, 0, 0), ξ³ = (0, 0, 1, 1, 0, 0), ξ⁴ = (0, 0, 0, 1, 1, 0) and ξ⁵ = (0, 0, 0, 0, 1, 1). For the choice of parameters I = 0.48, λ = 8, μ = 1.2, τ_r = 600 and U = 0.012 and for a range of noise amplitudes this matrix gives the following heteroclinic chain/latching dynamics:

Starting with initial condition close to ξ¹, the dynamics visits successively ξ², …, ξ⁵, the transition form ξⁱ to ξⁱ⁺¹ passing through the intermediate (not learned) state with only one excited neuron at rank i + 1, see Fig 7. Observe that as long as a variable x_j is “large” (close to 1) the corresponding synaptic variable s_j decreases until x_j comes close to 0. Then s_j increases, according to the time evolution driven by Eq (4).

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Fig 7. Chain of five patterns with 6 neurons and m = 2.

Panel (a) shows the firing rates x_j and panel (b) shows the synaptic variables s_j. Color code: blue = x₁, red = x₂, black = x₃, green = x₄, cyan = x₅, magenta = x₆. Same code for s_j.

https://doi.org/10.1371/journal.pone.0183710.g007

A case of a shared neuron with m = 3

This example gives a different option for the neural coding of items. Thus far the principle of our model has been that each item (e.g. the prime or target) is coded by a pattern of activity of all neurons in the network. As a consequence only one item can be ‘activated’ at a given time in a heteroclinic chain. The simplest case is when two patterns are activated in succession: the pattern coding for the prime followed by a pattern coding for the target [70, 72–74]. In this case either the prime or the target is ‘activated’ at a given time. Such priming mechanism can account for neuronal activities recorded in nonhuman primates in priming protocols where a prime is related to a single target. In that case priming relies on the successive activation of the presented prime and of the predicted target [15, 69]. However, in human studies priming is reported not only for targets directly related to the prime (Step 1 targets), but also for targets indirectly related to the prime through a sequence of one (Step 2 targets) or two (Step 3 targets) intermediate associates of the prime that are activated after the prime and before the target (e.g. [25] for a review). Such indirect priming has been accounted for by network models in which Step 1, Step 2 and Step 3 associates to the primes were coded by neural populations that can be activated simultaneously [25]. The present model of heteroclinic chains is of particular interest to account for the sequential activation of items involved in step priming. However, in priming studies the prime can still be reported by participants after processing of the target [117]. This suggests that the prime must be available in working memory at the end of the activation of the sequence of Step associates, that is neurons coding for the prime must be active at the end of the heteroclinic chain. The possibility to activated the prime in after several associates have been activated in a chain is no reproduced by models of priming based on latching dynamics [70, 71]. In the present model, a way to for neurons coding for the prime to be actvated at the end of the heteroclinic chain is simply to consider that a pattern (attractor state) does not correspond to a single item, but rather corresponds to several items each corresponding to the activity of a subgroup of neurons. In that case the activity of a neuron would correspond to the average activity of a population of neurons coding for and item [22]. Such population coding is consistent with recent models of priming in the cerebral cortex [25, 40, 118]. Intuitively, the first pattern in the chain codes for the prime only while the next pattern 2 in the chain codes for the combination of the prime and of the Step 1 target together, the pattern 3 codes for the Prime, Step1 target and Step 2 target, and so on. This way the population coding for the prime would be active throughout the entire computation.

In this section we present an example of a system of five neurons with three active neurons in each pattern and one neuron present in each pattern. For this we use the following connectivity matrix: (27) τ_r = 400, x_max = 1, U = 0.012. For this system, for the choice of the parameters I = 0.5, λ = 2.8 and μ = 1 we find by simulation a chain joining ξ₁ = (1, 1, 1, 0, 0), ξ₂ = (1, 0, 1, 1, 0), ξ₃ = (1, 0, 0, 1, 1), and . The time series of the solution is shown in Fig 8.

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Fig 8. Chain of three patterns with 5 neurons, m = 3 and one shared neuron.

Panel (a) shows the firing rates x_j and panel (b) shows the synaptic variables s_j Color code: blue = x₁, red = x₂, black = x₃, green = x₄, cyan = x₅. Same code for s_j.

https://doi.org/10.1371/journal.pone.0183710.g008

The above analysis of the network behavior shows that heteroclinic chains can develop in the case where a neuron (i.e. a population coding for the prime) is active for all the successive patterns in the chain. In other words the overlap between populations coding for different items is such that a subgroup of neurons coding for an item (e.g. the prime) can remain activated while another subgroup can be deactivated as the chain progresses. The network is able to keep previous stimuli activated (e.g. a prime) while at the same time it can activate a sequence of items (i.e. associates to the prime) that can be predicted on the basis of the prime. The compatibility between changing patterns in the chain and stable activity of a neuron or population of neurons allows to account for two fundamental properties exhibited by the brain, within a unified model. Due to the structure of the overlap in the coding of the memory items, this example combines the population coding used classically in models of priming in the cerebral cortex, in which a given item is coded by a given population of neurons, and the distributed coding used in Hopfield types models of priming, in which a given item is coded by the pattern of activity of all neurons in the network. In this way the present model aims at unifying our understanding of the coding of items in memory and of priming processes between these items.

Predictions on neural activities in priming

The present model makes predictions regarding the possibility for the prime to remain activated (remembered) or not (forgotten) as the chain progresses. In the network, activating patterns coding for several Step 1, Step 2, etc targets would make difficult the simultaneous persistent activity of the prime, due to retroactive interference based on inhibition generated by the ‘step’ targets [40]. The corresponding experimental prediction that could be tested in priming experiments is that the activity of neurons coding for the prime would decrease when successive targets are predicted in memory even though they are not actually presented. This could be visible in nonhuman primates on a decrease of the retrospective activity of neurons coding for the prime when a series of ‘Step’ targets is predicted. The behavioral counterpart in humans would be a decrease in the reportability of the prime when the length of the sequence of targets to predict increases.

Asymmetry of priming and of synaptic efficacies

Brunel [106] recently pointed out the possibility that the optimal synaptic matrix depends on the constraint imposed on the network, either storing patterns as stable states or storing patterns to be activated in sequences. The present results show that heteroclinic chains are possible with symmetric matrices built through Hebbian learning and specify the necessary conditions for sequences to arise in the network. Although asymmetric connections can improve the ability of the network to activate sequences of patterns, they are not a necessary condition. However, the present results also show that the conditions for heteroclinic chains impose strong constraints on the structure of the synaptic matrix, suggesting that although symmetric weights can be optimal for storage capacity, they are not an optimal solution for the activation of sequences. If the network codes for a given pattern 1 at a given time, the activation of the next pattern 2 in a chain requires that two given neurons i and j activate each other or not depending on their state within each pattern. For example, i but not j can be activated in 1, and the opposite in 2. Hence for an optimal sequence 1 → 2, i should activate j but j should not activate i. The present results show that the symmetry of the weights can be compensated by the sparsity of the network at the expense of an increasing number of neurons necessary to code for the patterns. Even though asymmetric heteroclinic chains are possible with symmetric synaptic efficacies, further analysis of heteroclinic chains with asymmetric learning rules would bring new evidence on the specific role of asymmetric weights on the required level of sparsity and on the reliability of latching dynamics.