A decision-making Fokker-Planck model in computational neuroscience

Minimal models for the explanation of decision-making in computational neuroscience are based on the analysis of the evolution for the average firing rates of two interacting neuron populations. While these models typically lead to multi-stable scenario for the basic derived dynamical systems, noise is an important feature of the model taking into account finite-size effects and robustness of the decisions. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker-Planck partial differential equation. In particular, we discuss the existence, positivity and uniqueness for the solution of the stationary equation, as well as for the time evolving problem. Moreover, we prove convergence of the solution to the the stationary state representing the probability distribution of finding the neuron families in each of the decision states characterized by their average firing rates. Finally, we propose a numerical scheme allowing for simulations performed on the Fokker-Planck equation which are in agreement with those obtained recently by a moment method applied to the stochastic differential system. Our approach leads to a more detailed analytical and numerical study of this decision-making model in computational neuroscience

DOI 10.1007/s00422-007-0144-6 ORIGINAL PAPER Deterministic analysis of stochastic bifurcations in multi-stable neurodynamical systems

decision-making and bistable perception, involve multistable phenomena under the influence of noise. The role of noise in a multistable neurodynamical system can be formally treated within the Fokker–Planck framework. Nevertheless, because of the underlying nonlinearities, one usually considers numerical simulations of the stochastic differential equations describing the original system, which are time consuming. An alternative analytical approach involves the derivation of reduced deterministic differential equations for the moments of the distribution of the activity of the neuronal populations. The study of the reduced deterministic system avoids time consuming computations associated with the need to average over many trials. We apply this technique to describe multistable phenomena. We show that increasing the noise amplitude results in a shifting of the bifurcation structure of the system.