Optimized brute-force algorithms for the bifurcation analysis of a binary neural network model.

Area of Science:

• Computational Neuroscience
• Theoretical Neuroscience
• Complex Systems

Background:

• Bifurcation theory analyzes neural network dynamics based on neurophysiological parameters.
• Existing methods are limited to infinite-size models, continuous outputs, or continuous-time discrete neurons.
• Widely used discrete-unit, finite-size models lack thorough bifurcation analysis.

Purpose of the Study:

• Introduce algorithms for semianalytical bifurcation analysis of finite-size firing-rate neural networks with binary rates and discrete time.
• Enable study of small networks (tens of neurons) unsuitable for current statistical methods.
• Characterize the impact of network parameters on multistability, oscillations, and symmetry breaking.

Main Methods:

• Numerical brute-force search for stationary and oscillatory solutions.
• Derivation of analytical bifurcation structure using state-to-state transition probability matrix.
• Development of efficient algorithms for arbitrary and sparse connectivity matrices.

Main Results:

• Algorithms determine how network parameters affect multistability.
• Analysis reveals emergence and period of neural oscillations.
• Identified formation of spontaneous symmetry breaking in neural populations.
• Demonstrated applicability to sparse networks with a highly efficient algorithm.

Conclusions:

• The new algorithms provide a powerful tool for analyzing discrete-time, finite-size neural networks.
• This methodology advances understanding of neural population dynamics and emergent behaviors.
• The study offers a Python implementation for practical application in computational neuroscience.