The dynamics of neural networks are significantly influenced by the strength of connections between neurons. Strong connections often lead to increased coordination among neurons in the same circuit. However, studies have revealed that even in the presence of strong connections, neural circuits can exhibit asynchronous activity. This phenomenon is observed experimentally and theoretically in sparse excitatory-inhibitory networks with a large number of neurons (N) and a limited number of connections (K << N), where the coupling strength is set as 1/sqrt(K) [1][2]. Hansel and Mato (2003) [3] investigated the scenario in which the asynchronous state existing in a neural network comprising two populations with extensively connected neurons (one excitatory and the other inhibitory) becomes unstable. The main goal of our project is to reproduce the phase diagram constructed by Hansel and Mato, for gaining a comprehensive understanding of the dynamical system properties of the network.

Our goal is to construct a computational model for studying networks comprising subpopulations of excitatory (E) and inhibitory (I) neurons. To express the coupling strength G^syn_ij,alpha,beta between neuron i in population alpha (alpha = E, I) and neuron j in population beta (beta = E, I), we will consider that the coupling only depends on the types of neurons (I/E) involved in the connection.In the case of this extensively connected network, the synaptic strengths are varied in inverse proportion to the system size.The normalized synaptic strengths, g_alpha,beta, are independent of the N_alpha [4] .

To emulate the intrinsic dynamics of a broad range of neurons, we employ quadratic integrate-and-fire (QIF) dynamics. This choice is motivated by its ability to serve as a reduced model near the firing onset for various type I conductance-based models, such as the Wang-Buszaki (WB) model. Our aim is to identify the parameters in the WB model that render the QIF model a robust approximation. We assign an external current that varies neuron to neuron, distributed according to a Gaussian distribution.

Within these systems, our focus is on analyzing the stability of the asynchronous state based on the trace and determinant of the coupling matrix. We anticipate observing various transitions from the asynchronous state (AS). For instance, a Saddle-Node bifurcation may lead to a different state characterized by a distinct firing rate. Alternatively, a Hopf bifurcation could bring about changes in oscillatory synchronous states. By examining these transitions, we aim to gain insights into the dynamic behavior of the neural network, furthering our understanding of how different bifurcations contribute to the network’s overall stability and firing patterns. All the simulations will be developed in Python.

References:

[1] C. van Vreeswijk and H. Sompolinsky. Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science, 274(5293):1724–1726, December 1996.

[2] Jeremie Barral and Alex D Reyes. Synaptic scaling rule preserves excitatory–inhibitory balance and salient neuronal network dynamics. Nature Neuroscience, 19(12):1690–1696, October 2016.

[3] D. Hansel and G. Mato. Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons. Neural Computation, 15(1):1–56, January 2003.

[4] D. Hansel and H. Sompolinsky. Chaos and synchrony in a model of a hypercolumn in visual cortex. Journal of Computational Neuroscience, 3(1):7–34, March 1996.