Laboratoire Matière et Systèmes Complexes

The emergence of collective behaviors of neuronal networks experimentally observed by the group of E. Moses in the Weizmann institute suggested a link with critical phenomena and led to the elaboration of a model of directed percolation on a graph : Each neuron is modelled by a two state system (either at rest or active) whose activation is determined by a threshold (quorum). A neuron becomes active when it has integrated a quorum m of signal sent by its active neighbors ; as soon as it becomes active, it sends in its turn signals to its neighbors. Starting at time t=0 from an initial state of the network where a given fraction f of randomly chosen nodes is set active, information spreads through the network and the process converges towards a stationary state characterized by the fraction of active neurons. The behavior of \phi as a function of the control parameter m and f is depicted by a phase diagram close to the one obtained in the framework of the Ising model under an external field. The most striking feature of such a diagram is a jump in \phi below a critical value m_c which indicates a collective behavior, understood as the appearance of a giant cluster. As a main result, a study of the critical behavior enabled to deduce that the connectivity of such neuronal networks is consistent with a random distribution of the incoming links well described by a Gaussian with a mean of about 50 to 100 links per neuron.

Propagation of the information with m=3

An example of network

Phase diagram of the quorum percolation model.

• We extended the validity limits of this model by taking into account the effect of memory when the decay characteristic time is no longer negligible compared to the propagation time between two neurons (the membrane of a neuron behaves as a capacitor continuously leaking ions). In the framework of a mean field approach, we were able to write down an equation describing the stochastic process ; we showed that explicit Monte Carlo simulations converge very quickly towards the mean field behavior as the network size increases for Gaussian distributions of incoming links. Furthermore, we showed that the decay alters the occurrence of a giant cluster by replacing the discontinuities in \phi by steep but finite slopes at the same time as it reduces the size of the apparent discontinuity. At last we gave strong numerical evidence to conjecture that even an infinitesimal decay destroys the critical point in the thermodynamical limit 1.

• A deep study of the critical behavior led us to develop an extension of the percolation model to continuous values of the quorum by introducing special functions. In the framework of a mean field approach, we obtained high precision values of the behavior of the critical point m_c with the mean and width of the Gaussian probability distribution of incoming links ; moreover, we showed that the very high non linearity of the self-consistent equation leads to numerical calculations of the critical exponent which slightly deviate from the ½ value expected from mean-field theory 2.

  

  Schematic representation of QP with decay. with m=3. Random disintegration of signals (decay) between steps (glowing links) changes the final state of the network ; without decay, the five neurons would be set active at time t=3

• An additional feature that we introduce in the model with respect to the original quorum percolation one is a distribution of activation thresholds : While in the original model the quorum m takes on the same value whatever the node may be, disorder on the quorum is introduced by randomly allocating to each node an integer value mi according to some Gaussian probability distribution independently of the network properties. mi accounts for the excitability of the node i In the framework of a mean-field approach, we were able to write a self-consistency equation whose qualitative behavior is close to the one obtained without disorder. A good agreement between Monte-Carlo simulations and the mean field approach is obtained in the connectivity range of neuronal networks (a Gaussian distribution with an average from 25 to 300 incoming links) as soon as the size of the network is great enough (about 20 000 nodes). Furthermore, we showed the occurrence of disorder independent fixed points in the subcritical region.

  

  Effects of the disorder (width of the thresholds distribution m) on the occurrence of the giant cluster : As the disorder increases (from right to left), the position of the jump is shifted towards low values of f while the size the giant cluster decreases until the disorder is strong enough to destroy the percolation. Note the agreement between Monte Carlo simulation (colored points) and the mean field results (black points).

Disorder independent fixed points in the subcritical region : For each value of the mean excitability, m is increased (from the red to the blue points).