71,695 research outputs found
Divisive Normalization from Wilson-Cowan Dynamics
Divisive Normalization and the Wilson-Cowan equations are influential models
of neural interaction and saturation [Carandini and Heeger Nat.Rev.Neurosci.
2012; Wilson and Cowan Kybernetik 1973]. However, they have not been
analytically related yet. In this work we show that Divisive Normalization can
be obtained from the Wilson-Cowan model. Specifically, assuming that Divisive
Normalization is the steady state solution of the Wilson-Cowan differential
equation, we find that the kernel that controls neural interactions in Divisive
Normalization depends on the Wilson-Cowan kernel but also has a
signal-dependent contribution. A standard stability analysis of a Wilson-Cowan
model with the parameters obtained from our relation shows that the Divisive
Normalization solution is a stable node. This stability demonstrates the
consistency of our steady state assumption, and is in line with the
straightforward use of Divisive Normalization with time-varying stimuli.
The proposed theory provides a physiological foundation (a relation to a
dynamical network with fixed wiring among neurons) for the functional
suggestions that have been done on the need of signal-dependent Divisive
Normalization [e.g. in Coen-Cagli et al., PLoS Comp.Biol. 2012]. Moreover, this
theory explains the modifications that had to be introduced ad-hoc in Gaussian
kernels of Divisive Normalization in [Martinez et al. Front. Neurosci. 2019] to
reproduce contrast responses. The proposed relation implies that the
Wilson-Cowan dynamics also reproduces visual masking and subjective image
distortion metrics, which up to now had been mainly explained via Divisive
Normalization. Finally, this relation allows to apply to Divisive Normalization
the methods which up to now had been developed for dynamical systems such as
Wilson-Cowan networks
Physiologically motivated multiplex Kuramoto model describes phase diagram of cortical activity
We derive a two-layer multiplex Kuramoto model from weakly coupled
Wilson-Cowan oscillators on a cortical network with inhibitory synaptic time
delays. Depending on the coupling strength and a phase shift parameter, related
to cerebral blood flow and GABA concentration, respectively, we numerically
identify three macroscopic phases: unsynchronized, synchronized, and chaotic
dynamics. These correspond to physiological background-, epileptic seizure-,
and resting-state cortical activity, respectively. We also observe frequency
suppression at the transition from resting-state to seizure activity.Comment: 8 pages, 3 figure
Critical behaviour of the stochastic Wilson-Cowan model
Spontaneous brain activity is characterized by bursts and avalanche-like dynamics, with
scale-free features typical of critical behaviour. The stochastic version of the celebrated Wilson-
Cowan model has been widely studied as a system of spiking neurons reproducing
non-trivial features of the neural activity, from avalanche dynamics to oscillatory behaviours.
However, to what extent such phenomena are related to the presence of a genuine critical
point remains elusive. Here we address this central issue, providing analytical results in the
linear approximation and extensive numerical analysis. In particular, we present results supporting
the existence of a bona fide critical point, where a second-order-like phase transition
occurs, characterized by scale-free avalanche dynamics, scaling with the system size and a
diverging relaxation time-scale. Moreover, our study shows that the observed critical behaviour
falls within the universality class of the mean-field branching process, where the exponents
of the avalanche size and duration distributions are, respectively, 3/2 and 2. We also
provide an accurate analysis of the system behaviour as a function of the total number of
neurons, focusing on the time correlation functions of the firing rate in a wide range of the
parameter space
Modeling focal epileptic activity in the Wilson-Cowan model with depolarization block
Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity driven depolarization block, i.e. they saturate. We examined the effect of such a saturation in the Wilson–Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function. We discuss experimental recordings during a seizure that support this substitution. Next we perform a bifurcation analysis on the Wilson–Cowan model with a Gaussian activation function. The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity. Analysis of coupled local networks then shows that such high activity can stay localized or spread. Specifically, in a spatial continuum we show a wavefront with inhibition leading followed by excitatory activity. We relate our model simulations to observations of spreading activity during seizures
Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition
Fifty years ago, Wilson and Cowan developed a mathematical model to describe
the activity of neural populations. In this seminal work, they divided the
cells in three groups: active, sensitive and refractory, and obtained a
dynamical system to describe the evolution of the average firing rates of the
populations. In the present work, we investigate the impact of the often
neglected refractory state and show that taking it into account can introduce
new dynamics. Starting from a continuous-time Markov chain, we perform a
rigorous derivation of a mean-field model that includes the refractory
fractions of populations as dynamical variables. Then, we perform bifurcation
analysis to explain the occurance of periodic solutions in cases where the
classical Wilson-Cowan does not predict oscillations. We also show that our
mean-field model is able to predict chaotic behavior in the dynamics of
networks with as little as two populations.Comment: 14 pages, 14 figure
- …