Dynamic branching in a neural network model for probabilistic prediction of sequences

Abstract

An important function of the brain is to predict which stimulus is likely to occur based on the perceived cues. The present research studied the branching behavior of a computational network model of populations of excitatory and inhibitory neurons, both analytically and through simulations. Results show how synaptic efficacy, retroactive inhibition and short-term synaptic depression determine the dynamics of selection between different branches predicting sequences of stimuli of different probabilities. Further results show that changes in the probability of the different predictions depend on variations of neuronal gain. Such variations allow the network to optimize the probability of its predictions to changing probabilities of the sequences without changing synaptic efficacy.

Appendix: The latching dynamics

1.1 A.1 Sketch of the analysis in Köksal Ersöz et al. (2020)

A sequence of $N-1$ learned patterns $\xi ^i=(\xi ^i_1\dots ,\xi ^i_{N-1})$ was considered in a network of N units, where $\xi ^k_i=\xi ^k_{i+1}=1$ and the other coordinates are 0. These patterns sit on vertices of the hypercube $[0,1]^N$ and the trajectories of the activity Eq. (5) lie within the hypercube. The analysis will be simplified by the presence of the factor $x_i(1-x_i)$ in (5), which forces the trajectories starting on one face of the hypercube to lie within that face, whatever the dimension of the face. In particular, the vertices are the steady-states of (5). The evolution of the system is governed by the activity Eq. (5) and the STD is driven by (1). The corresponding connectivity matrix is (7), according to the learning rule (3). In the following we sketch without proofs the analysis in (Köksal Ersöz et al., 2020).

The first question is about the stability of learned patterns $\xi ^i$. This is determined by the sign of the eigenvalues at $\xi ^i$ of the Jacobian matrix of the system (5), which we write $\sigma ^i_j$ ($j=1,\dots ,N$). These eigenvalues are easily computed because the edges emanating from $\xi ^i$ on the hypercube are the eigendirections. It can be seen that all eigenvalues with $j<i$ or $j>i+1$ are negative if the condition $2\lambda +I>S$ is satisfied ($S=(1+\rho )^{-1}<1$ is the value towards which the synaptic strength $s_i$ decays as time elapses). The two remaining eigenvalues are

$$\begin{aligned} \begin{aligned}\sigma ^i_i&=\mu +2\lambda +I-ms_i-s_{i+1}\\\sigma ^i_{i+1}&=\mu +2\lambda +I-s_i-m's_{i+1} \end{aligned} \end{aligned}$$

(11)

where $m=1$ if $i=1$, $m'=1$ if $i=N-1$, and $m=m'=2$ for all other values of i.

Without STD the learned patterns $\xi ^i$ with $1<i<N-1$ are stable under the mild condition $S<2\lambda +I<3-\mu$. When $i=1$ or $N-1$ the condition becomes $S<2\lambda +I<2-\mu$, which is more restrictive. We assumed in Köksal Ersöz et al. (2020) that $I=0$. The results are still valid with $I>0$ small enough. In this study we have set $I=0$.

If STD is on, $s_i$ and $s_{i+1}$ decay towards $0<S<1$, so that eventually $\sigma ^i_i$ or $\sigma ^i_{i+1}$ may become positive at finite time. When $i=1$, (11) shows that $\sigma ^1_1>\sigma ^1_2$, therefore $\xi ^i$ becomes first unstable and the direction of this instability is $x_1$. When $i>1$, $m=m'$ in (11), so that $\sigma ^i_i$ and $\sigma ^i_{i+1}$ may become positive simultaneously. However, thanks to the previous switch $\xi ^{i-1}\rightarrow \xi ^i$, $s_i$ has been decaying towards S while $s_{i+1}$ was still close to 1 ($x_{i+1}=0$ at $\xi ^{i-1}$). Hence in any case $\xi ^i$ is first destabilized along $x_i$.

We claim that “typical” trajectories starting near $\xi ^i$ will converge towards $\xi ^{i+1}$ by first decreasing $x_i$ towards 0, so that the trajectory converges for some time towards the intermediate state $\hat{\xi }^i=(0,1,0,\dots ,0)$ (which represents the “overlap” between the learned patterns $\xi ^i$ and $\xi ^{i+1}$), then increasing $x_{i+2}$ towards 1, so that $\xi ^{i+1}$ is reached. Noise is an essential ingredient of this behavior as we will see. The proof of the claim relies on the following properties of the system: (i) the synaptic variables $s_j$ are slow compared to the activity variables $x_j$ and can be seen, in the limit of “stationary variables”, as bifurcation parameters for the $x_j$’s. This is the idea of “dynamic bifurcation”. These bifurcations drive the dynamics of the activity variables. (ii) The analysis can be restricted to the flow-invariant 2 dimensional face $F_i$ on the hypercube, with vertices $\xi ^{i}$, $\hat{\xi }^{i}$ and $\xi ^{i+1}$. In $F_i$, $x_{i+1}=1$, $x_j=0$ when $j<i$ or $j>i+2$ and the coordinates are $x_i$, $x_{i+2}$. In these coordinates $\xi ^i=(1,0)$, $\xi ^{i+1}=(0,1)$ and $\hat{\xi }^i=(0,0)$.

Without synaptic plasticity, the learned patterns are stable and the dynamics in $F_i$ looks like Fig. 9. When STD is on, synaptic strengths $s_i$ of active neurons diminish slowly, which triggers successive transitions in $F_i$. The proof of the claim is provided in Köksal Ersöz et al. (2020) and relies upon slow-fast dynamical systems theory. Here we only describe the possible scenarios.

Fig. 9

A typical phase portrait of the flow in the square $F_i$ in the absence of synaptic depression. Green spots correspond to the learned patterns. Other spots are unstable steady-states

The eigenvalue at $\xi ^i$ along coordinate $x_i$ is given in (11) and note that $\hat{\xi }^i$ has a double eigenvalue in $F_i$, given by

$$\begin{aligned} \hat{\sigma }^i_i=\hat{\sigma }^i_{i+2}=-\lambda +s_{i+1} \end{aligned}$$

(12)

Initially, $\xi ^i$ is stable. Let $t_{(1,0)}$ be the time at which $ms_i(t_{(1,0)})+s_{i+1}(t_{(1,0)})=\mu +2\lambda$ so that $\sigma ^i_i$ becomes positive, and let $t_{0,0}$ be the time at which the initially positive eigenvalue $\hat{\sigma }^i_i$ becomes negative. This happens when $s_{i+1}(t_{(0,0)})=\lambda$. Depending on the order of these two critical times two scenarios can occur.

Scenario 1

$t_{(1,0)}<t_{(0,0)}$. A (dynamic) bifurcation of a stable equilibrium occurs on the edge $x_{i+2}=0$ from the equilibrium (1, 0). This equilibrium travels towards (0, 0), which is reached at time $t_{(0,0)}$. A trajectory starting at initial time from the vicinity of $\xi ^i=(1,0)$ will therefore follow the edge $x_{i+2}=0$ until it arrives in a neighborhood of $\hat{\xi }^i$.

Scenario 2

$t_{(1,0)}>t_{(0,0)}$, so that an equilibrium bifurcates first from (0, 0) along the edge $x_{i+2}=0$. The point (1, 0) is still stable but its basin of attraction shrinks. Then sufficient noise may allow the trajectory starting in the vicinity of (1, 0) to escape its basin of attraction and converge towards $\hat{\xi }^i$.

In both scenarios, in order for the trajectory to converge to $\xi ^{i+1}$, one must have $\sigma ^{i+1}_{i+2}<0$, which by (11) requires $s_{i+1}+m's_{i+2}>\mu +2\lambda$. Since $s_{i+2}$ is still close to 1 this condition is easily satisfied, as was seen above, unless $m'=1$. In this case $i=N-1$, which implies that the last pattern of the chain can hardly be attained by the system.

Additional constraints shared by the two scenarios must be satisfied, notably that $\hat{\xi }^i$ be stable in the transverse directions to $F_i$. This gives the condition $\mu <\lambda$ (see Fig. 10).

Fig. 10

Snapshots of the transition $\xi ^i\rightarrow \xi ^{i+1}$ in the square $F_i$, in the scenarios 1 and 2. Bifurcated equilibria are blue. Red lines mark the boundary of the basin of attraction of $\xi ^{i+1}$ and dashed circles mark the closest distance for a possible stochastic jump over it. Scenario 1: a $\xi ^i$ is still stable. b Trajectory starting near $\xi ^i$ follows the saddle which has dynamically bifurcated on the edge $x_{i+2=0}$, until it reaches $\hat{\xi }^i$. Then arbitrary small noise can suffice to allow trajectory to jump towards $\xi ^{i+1}$. c $\hat{\xi }^i$ has become stable by dynamic bifurcation of a saddle point on the edge $x_{i+2=0}$, but trajectory can jump over its basin of attraction by effect of sufficiently strong noise and then converge to $\xi ^{i+1}$. Scenario 2: a same as scenario 1. b A saddle has bifurcated on the edge $x_{i+2}$, so that a trajectory starting near $\xi ^i$ can only converge to $\xi ^{i+1}$ by the effect of strong enough noise. c Eventually the trajectory can converge towards $\hat{\xi }^i$ and the same situation as in scenario 1 holds

1.2 A.2 Latching dynamics in the case of n-ways branching

We analyze how the presence of a n-ways branching node in the graph of learned patterns affects the occurrence of latching dynamics. For the sake of clarity we concentrate on the example associated with the connectivity matrix (8), where the node $\xi ^2$ has multiplicity 3 (Example 1, Sect. 2.2). The generalization to higher multiplicity is straightforward.

Now the equations for the neural activity are (10) where the term $\nu _i$ is 0 if $i\ne 3$ and $\nu _3=\lambda$. Along chains which do not encounter the branching “choice” $\xi ^2\rightarrow \xi ^3$ or $\xi ^2\rightarrow \xi ^5$, the latching dynamics analysis proceeds as sketched in A.1. Indeed the higher excitation at unit 3, which in the factor of $x_i(1-x_i)$ in (10) reads $3s_3x_3$, is compensated by the higher inhibition $-3\lambda x_3$. We therefore concentrate on the piece of the graph defined by the multi node $\xi ^2$ and its adjacent patterns $\xi ^3$ and $\xi ^5$.

We expect $\xi ^2$ becomes first unstable along its coordinate $x_2$, converging towards the intermediate state $\hat{\xi }^2=(0,0,1,0,\dots ,0)$. But then either $x_4$ or $x_6$ should rise to 1 (transitions $\xi ^2\rightarrow \xi ^3$ or $\xi ^2\rightarrow \xi ^5$. Therefore we need to consider the cube $F_3$ defined by setting $x_3=1$ and $x_j=0$ for $j\ne 2,4,6$. In $F_3$, $\xi ^2=(1,0,0)$, $\xi ^3=(0,1,0)$, $\xi ^5=(0,0,1)$ and $\hat{\xi }^2=(0,0,0)$. The equations in $F_3$ are

$$\begin{aligned} \begin{aligned} \dot{x_2} =&\ x_2(1-x_2)(-\mu x_2 -\lambda (x_2+x_4+x_6+1)\\&+ 2s_2x_2+s_3) \\ \dot{x_4} =&\ x_4(1-x_4)(-\mu x_4 -\lambda (x_2+x_4+x_6+1) \\&+ s_3+2s_4x_4) \\ \dot{x_6} =&\ x_6(1-x_6)(-\mu x_6 -\lambda (x_2+x_4+x_6+1) \\&+ s_3+2s_6x_6) \end{aligned} \end{aligned}$$

(13)

Observe that restricting further the equations to the squares defined by $x_4=0$ or $x_6=0$ leads to systems completely analogous to those in $F_i$ in A.1, so that the same analysis applies to both faces. Moreover the transverse eigenvalues to these faces at $\xi ^3$ and $\xi ^5$ are, in the notations of A.1, $\sigma ^3_6=\sigma ^5_4=-2\lambda +s_3<0$. Therefore the dynamics will closely follow one of the faces and the transitions $\xi ^2\rightarrow \xi ^3$ and $\xi ^2\rightarrow \xi ^5$ will occur with equal probability. Figure 11 illustrates the global dynamics in $F_3$ in a case corresponding to scenarios 1-(c) or 2-(c) in Fig. 10.

Fig. 11

A sketch of the dynamics in $F_3$ corresponding to Fig. 10, scenario 1-(c) or scenario 2-(c)

1.3 A.3 Relation between the number of excitatory connections and self-inhibition

Consider that the system is in $F_3$ and $(x_2, x_4, x_6) = (1,0,0)$, which corresponds to $\xi ^2$. As $s_i$ variables varies in slow time, the system (13) will undergo several bifurcations that leading to either $\xi ^2 \rightarrow \hat{\xi }^2 \rightarrow \xi ^3$ or $\xi ^2 \rightarrow \hat{\xi }^2 \rightarrow \xi ^5$ transitions, with equal probabilities. The conditions for the $\xi ^2 \rightarrow \hat{\xi }^2 \rightarrow \xi ^3$ transition can be derived form the eigenvalue expressions:

• Initially stable equilibrium point $(x_2, x_4, x_6) = (1,0,0)$ becomes unstable at $t = t^3_{(1, 0, 0)}$:

  $$2 s_2(t^3_{(1, 0, 0)}) + s_3(t^3_{(1, 0, 0)}) = \mu + 2\lambda .$$

• Initially unstable equilibrium point $(x_2, x_4, x_6) = (0,0,0)$ becomes stable at $t = t^3_{(0, 0, 0)}$:

  $$s_3(t^3_{(0, 0, 0)}) = \lambda .$$

The next transition is $\xi ^3 \rightarrow \hat{\xi }^3 \rightarrow \xi ^4$ in $F_4$ defined by setting $x_4 = 1$ and $x_j = 0$ for $j\ne 3,5$. In $F_4$ the equations of the neural activity are reduced to

$$\begin{aligned} \begin{aligned} \dot{x_3} =&\ x_3(1-x_3)(-\mu x_3 -\lambda (x_3+x_5+1)\\&-\nu _3x_3+ 3s_3x_3+s_4)\\ \dot{x_5} =&\ x_5(1-x_5)(-\mu x_5 -\lambda (x_3+x_5+1)\\&-\nu _5x_5 +s_4+s_5x_5) \end{aligned} \end{aligned}$$

(14)

The conditions for the transition $\xi ^3 \rightarrow \hat{\xi }^3 \rightarrow \xi ^4$ are

• Initially stable equilibrium point $(x_3, x_5) = (1,0)$ becomes unstable at $t = t^4_{(1, 0)}$:

  $$3 s_3(t^4_{(1, 0)}) + s_4(t^4_{(1, 0)}) = \mu + 2\lambda + \nu _3.$$

• Initially unstable equilibrium point $(x_3, x_5) = (0,0)$ becomes stable at $t = t^4_{(0, 0)}$:

  $$s_4(t^4_{(0, 0)}) = \lambda .$$

Assume that $t^3_{(1, 0, 0)} < t^3_{(0, 0, 0)}$ and $t^4_{(1, 0)} < t^4_{(0, 0)}$. To have equally distributed pattern duration along a sequence, the $s_i$ values at the bifurcation moments should read $s_2(t^3_{(1, 0, 0)}) = s_3(t^4_{(1, 0)})$ and $s_3(t^3_{(1, 0, 0)}) = s_4(t^4_{(1, 0)})$. With this restriction, the following algebraic condition can be deduced:

$$\begin{aligned} \begin{aligned} 2 s_3(t^4_{(1, 0)}) + s_4(t^4_{(1, 0)})= & {} \mu + 2\lambda ,\\ 3 s_3(t^4_{(1, 0)}) + s_4(t^4_{(1, 0)})= & {} \mu + 2\lambda + \nu _3, \end{aligned} \end{aligned}$$

that is

$$\begin{aligned} s_3(t^4_{(1, 0)})= & {} \nu _3. \end{aligned}$$

More generally, the relation between the number excitatory connections $d_i$ and the level of self-inhibition $\nu _i$ is given by:

$$\begin{aligned} (d_i-2) s_i (t_{(1, 0)})= & {} \nu _i, \end{aligned}$$

with $s_i (t_{(1, 0)})$ being the value of $s_i$ at the moment where a learned pattern changes stability. This implies that to succeed in sequential activation between the learned patterns, over-excitation imposed by the learning should be balanced by self-inhibition.