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Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. (English) Zbl 1131.92016
Summary: Computational techniques within the population density function (PDF) framework have provided time-saving alternatives to classical Monte Carlo simulations of neural network activity. Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases. In a detailed theoretical and computational study, we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method [D. Cai et al., Commun. Math. Sci. 4, No. 1, 97–127 (2006; Zbl 1107.82037)] as applied to integrate-and-fire neurons that receive excitatory synaptic input only. When the unitary postsynaptic conductance event has a single-exponential time course, the evolution equation for the PDF is a partial differential integral equation in two state variables, voltage and excitatory conductance. In the moment closure method, one approximates the conditional $$k$$ th centered moment of excitatory conductance given voltage by the corresponding unconditioned moment. The result is a system of $$k$$ coupled partial differential equations with one state variable, voltage, and k coupled ordinary differential equations.
Moment closure at $$k=2$$ works well, and at $$k=3$$ works even better, in the regime of high dynamically varying synaptic input rates. Both closures break down at lower synaptic input rates. Phase-plane analysis of the $$k=2$$ problem with typical parameters proves, and reveals why, no steady-state solutions exist below a synaptic input rate that gives a firing rate of 59 s$$^{-1}$$ in the full 2D problem. Closure at $$k=3$$ fails for similar reasons. Low firing-rate solutions can be obtained only with parameters for the amplitude or kinetics (or both) of the unitary postsynaptic conductance event that are on the edge of the physiological range. We conclude that this dimension-reduction method gives ill-posed problems for a wide range of physiological parameters, and we suggest future directions.

##### MSC:
 92C20 Neural biology 35Q92 PDEs in connection with biology, chemistry and other natural sciences 37N25 Dynamical systems in biology
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