Wei Qin,
Andre Peterson
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https://unimelb.zoom.us/u/adDa1aiD6C Phase transitions are often used to describe pathological brain state transitions observed in neurological diseases such as epilepsy. Typically, the study of the dynamics of neurons that are nonlinearly coupled and have complex network structures is done via large scale numerical simulations, which are mathematically intractable. Otherwise, analysis is performed where the network structure is averaged over and made spatially homogeneous. For a networked nonlinear dynamical system, phase transitions or bifurcations are computed via changes in the local stability around the fixed points. However, in such a system it is very difficult to compute the fixed points as the dimensionality of the system becomes large due to nested nonlinearities. We know from numerical simulations that the system becomes `chaotic`[1] as the order parameter (variance of the connectivity matrix) is increased and that microscopically this phase transition corresponds to an exponential increase in the number of fixed points[3]. This phase transition has also been computed for heterogeneous network structures such as Dale’s law[2]. However, it is very difficult to numerically verify these results. To quantify the change in network dynamics, we compute the entropy, a quantity which describes the number of states or information in a system. We show in this paper that the Network entropy (NE), a term derived from Shannon Entropy, can be used as a numerical indicator of a change in the number of equilibria. Hence, it is also a numerical method to estimate the change in stability of a network. It is developed via a Symbolic Dynamic approach based on probability distributions of the system state, which provides a measure of the number of states of the system.
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In this paper, a first order neural model with a time-constant and instantaneous synapses is networked. The network connectivities are described by a random matrix with mean and variance. Dale’s law can be integrated into the model by changing the connectivity matrix. We estimate the stability of a network via measuring the entropy of the network states using numerical simulations with different realisations of the connectivity matrix. The result demonstrates that the transition points from the analytical results for each case coincided with the measured NEs. It suggests the NEs can be used in numerical simulations to estimate the changes in the number of the fixed points, a.k.a. the phase transitions. This work provides a novel approach to estimate the network states and phase transitions via numerical simulations. Future works are needed to discover the mathematical relationship between the fixed points and entropy. Furthermore, it is interesting to use entropy to predict the dynamical behaviours of a system in an early stage. The discovery can be used to understand the brain state transitions and for the early diagnosis of neurological diseases, such as epilepsy.
References
1\. Stern M, Sompolinsky H, Abbott LF. Dynamics of random neural networks with bistable units. Phys.l Rev. E. 2014;90(6):062710.
2\. Ipsen JR, Peterson AD. Consequences of Dale's law on the stability- complexity relationship of random neural networks. 2019;arXiv:1907.07293.
3\. Wainrib G, Touboul J. Topological and dynamical complexity of random neural networks. Phys. rev. letters. 2013;110(11):118101.
