Abstract
Mean-field models of the brain approximate spiking dynamics by assuming that each neuron responds to its neighbors via a naive spatial average that neglects local fluctuations and correlations in firing activity. In this paper we address this issue by introducing a rigorous formalism to enable spatial coarse-graining of spiking dynamics, scaling from the microscopic level of a single type 1 (integrator) neuron to a macroscopic assembly of spiking neurons that are interconnected by chemical synapses and nearest-neighbor gap junctions. Spiking behavior at the single-neuron scale ℓ≈10μm is described by Wilson's two-variable conductance-based equations [H. R. Wilson, J. Theor. Biol. 200, 375 (1999)], driven by fields of incoming neural activity from neighboring neurons. We map these equations to a coarser spatial resolution of grid length Bℓ, with B≫1 being the blocking ratio linking micro and macro scales. Our method systematically eliminates high-frequency (short-wavelength) spatial modes q(->) in favor of low-frequency spatial modes Q(->) using an adiabatic elimination procedure that has been shown to be equivalent to the path-integral coarse graining applied to renormalization group theory of critical phenomena. This bottom-up neural regridding allows us to track the percolation of synaptic and ion-channel noise from the single neuron up to the scale of macroscopic population-average variables. Anticipated applications of neural regridding include extraction of the current-to-firing-rate transfer function, investigation of fluctuation criticality near phase-transition tipping points, determination of spatial scaling laws for avalanche events, and prediction of the spatial extent of self-organized macrocolumnar structures. As a first-order exemplar of the method, we recover nonlinear corrections for a coarse-grained Wilson spiking neuron embedded in a network of identical diffusively coupled neurons whose chemical synapses have been disabled. Intriguingly, we find that reblocking transforms the original type 1 Wilson integrator into a type 2 resonator whose spike-rate transfer function exhibits abrupt spiking onset with near-vertical takeoff and chaotic dynamics just above threshold.
Highlights
Effective modeling of brain dynamics requires assimilation of activity over multiple scales
Synaptic and ion-channel noise can have a significant impact on neuron dynamics, and this has stimulated investigation of stochastic adaptations of these model equations using both additive noise and Markov-chain approaches [6,7,8,9,10,11]
We present a scheme to scale systematically the dynamics of a single spiking neuron to the bulk environment of an assembly of such neurons
Summary
Effective modeling of brain dynamics requires assimilation of activity over multiple scales. The process of spike generation is governed by voltagedependent changes in Na+ and K+ ion-channel conductances that shape the growth, deceleration, and subsequent recovery of the membrane voltage. Synaptic and ion-channel noise can have a significant impact on neuron dynamics, and this has stimulated investigation of stochastic adaptations of these model equations using both additive noise and Markov-chain approaches [6,7,8,9,10,11].
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