Lotka-Volterra Models Describe Large-Scale Activity of Balanced Random Networks

The large-scale dynamics of a balanced random network of excitatory and inhibitory integrate-and-fire neurons is the focus of our study. Based on the dynamical equations of the model, a mean field approach was employed to reduce the dimensionality of the network dynamics [1,2]. We analyzed the joint activity dynamics of excitatory and inhibitory populations using a pair of mutually interacting differential equations. In absence of a voltage leak for individual neurons, and for negligible synaptic transmission delay, these equations take the form of Lotka-Volterra equations. These are known for describing predator-prey systems, which correspond to excitatory and inhibitory populations in our case. We tried to find optimal parameters for the non-autonomous differential equations given a dataset from a numerical simulations of a network. Moreover, we attempted to analytically infer the parameters and compare it with the statistical estimates.

As a next step, we analyzed the stability of the network considering two bifurcation parameters: “g”, the relative strength of recurrent inhibition, which controls the balance between excitation and inhibition in the network, and “eta”, the intensity of external input to the network. We found out that for a value of “g” that keeps the exact balance between excitation and inhibition, a bifurcation from unstable to stable network dynamics takes place. This bifurcation separates Synchronous Regular (SR) from Asynchronous Irregular (AI) activity of the network, similar to what was found in a previous study on the same network using a Fokker-Planck approach [3].

It has been shown that Lotka-Volterra equations are capable of representing switching dynamics between different states of neural networks [4]. Our analysis represents a first step toward analyzing the dynamics of more complex “networks of networks” that are implicated in various cognitive abilities of the brain.

References
1. Cardanobile S, Rotter S. Multiplicatively interacting point processes and applications to neural modeling. Journal of Computational Neuroscience 28(2): 267-284, 2010
2. Cardanobile S, Rotter S. Emergent properties of interacting populations of spiking neurons. Frontiers in Computational Neuroscience 5: 59, 2011
3. Brunel N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience 8(3): 183-208, 2000Bick C, Rabinovich M. On the occurrence of stable heteroclinic channels in Lotka-Volterra models. Dynamical Systems 25: 97-110, 2010
4. Bick C, Rabinovich M. On the occurrence of stable heteroclinic channels in Lotka-Volterra models. Dynamical Systems 25: 97-110, 2010

Support by the German Federal Ministry of Education and Research (BMBF; grant 01GQ0420 to BCCN Freiburg) is gratefully acknowledged.