From individual spiking neurons to population behavior: Systematic elimination of short-wavelength spatial modes

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 $\ell \approx 10\mu \mathrm{m}$ 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 $B\ell$, with $B\gg 1$ being the blocking ratio linking micro and macro scales. Our method systematically eliminates high-frequency (short-wavelength) spatial modes $\vec{q}$ in favor of low-frequency spatial modes $\vec{Q}$ 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.

Figure 1

Mapping of cortical field equations to a coarser spatial scale. (a) Continuum field equations for excitable cortical tissue are discretized to a fine-grained micro-scale lattice of spiking neurons (b) [see Fig. 4], then reblocked onto a coarser spatial grid (c) [Fig. 4] before inversion back to continuum form (d). The result is a set of spiking-neuron field equations describing spatiotemporal dynamics at the coarser spatial scale.

Figure 2

Spiking activity in upstream neuron $a$ leads to perturbation in the soma voltage of downstream neuron $b$. The flow of activity is mediated by presynaptic flux ${\varphi }_{ab}$ which transforms to postsynaptic flux ${\mathrm{\Phi }}_{ab}$ on crossing the synaptic interface between transmitting axon and receiving dendrite.

Figure 3

Cytoplasmic potassium-ion currents flow between cell interiors via open gap-junction connections whose conductance is $g^{\text{gap}}$. ${\mathrm{K}}^{+}$ will accumulate at the $b\text{th}$ neuron at $(x,y)$ if the incoming ionic flux $(I_1+I_3)$ exceeds the outgoing flux $(I_2+I_4)$, leading to a local increase in potassium concentration, boosting recovery $R_b$ by raising transmembrane ${\mathrm{K}}^{+}$-channel conductance and hence ${\mathrm{K}}^{+}$ efflux into the extracellular space during spiking. Spatial variability in $I_b^{\text{diff}}$ will drive diffusion in the recovery variable, i.e., $R_b^{\text{diff}}\sim {\nabla }^2{I}_b^{\text{diff}}$.

Figure 4

Spatial and Fourier representations of a gridded cortex. (a) Microcell lattice: A square patch of cortex of dimension $L_0\times L_0$, assumed periodic, is partitioned into a lattice of $(2n+1)\times (2n+1)$ “microcells,” each of side length $\ell$ (small red square) such that $L_0=(2n+1)\ell$. (b) Macrocell lattice: The cortical patch is reblocked into $(2N+1)\times (2N+1)$ “macrocells,” each of side length $L=B\ell$ (larger red square) so that $L_0=(2N+1)L$, with $B=(2n+1)/(2N+1)\gg 1$. (For illustrative purposes, we have set $B=3$ in this figure.) (c) Fourier lattice: In Fourier space, these griddings correspond to wave-number domains $|q_x|,|q_y|\le \pi /\ell$ for the finely spaced microgrid, and the much reduced domain $|Q_x|,|Q_y|\le \pi /L$ for the coarse macrogrid.

Figure 5

Upper limits for recovery diffusivity $D_R$ as a function of voltage diffusion coefficient $D_b$. The dashed curve corresponds to the small wave-number limit $q=Q^{max}=\pi /\left(B\ell \right)$, where $B=100$ is the blocking ratio, while the solid line applies the more conservative limit $q=q^{max}=\pi /\ell$. The circled point shows the selected operating point $D_b=6.60\times {10}^{-4}$ ${\mathrm{m}}^2$/s, $D_R=2.47\times {10}^{-13}$ ${\mathrm{m}}^4$/V.

Figure 6

Current-to-firing rate transfer functions for (a) standard Wilson integrator neuron [Eqs. ((1), (2))] (small black circles) and (b) reblocked diffusive Wilson neuron [Eqs. (73) and (74)] (red asterisks). The transfer function for the standard neuron increases smoothly from zero, whereas the reblocked neuron shows a discontinuity with near-vertical takeoff above threshold. The threshold currents are $I_{\text{dc}}^{\text{crit}}/\left(\text{A}{\text{m}}^{-2}\right)\approx 0.21475\text{and}0.34652$ for the standard and reblocked neurons, respectively. The large blue circles mark the $I_{\text{dc}}$ drive-current values selected for the six time series displayed in Fig. 7.

Figure 7

Comparison between firing patterns for (a) standard Wilson integrator neuron (left-hand panels) and (b) reblocked diffusive Wilson neuron (right-hand panels) for three settings of $I_{\text{dc}}$ drive current just above threshold: $I_{\text{dc}}=I_{\text{dc}}^{\text{crit}}+\delta I$. For the reblocked neuron in (b), firing onset is abrupt and chaotic, and exhibits damped nonlinear oscillations between firing events, whereas for the the standard integrator, firing onset is smooth and graduated. Evidently, reblocking has transformed the type 1 integrator into a type 2 resonator neuron.