Voltage-based Neural Field Equation

• Voltage-based NFE is a continuum model where membrane potentials evolve via a local leak term and spatially nonlocal synaptic integration.
• It bridges cellular biophysics and population dynamics by deriving from microscopic leaky integrate-and-fire models and incorporating dendritic processing and electrodynamic feedback.
• Analytical and numerical frameworks validate traveling waves, stability, and convergence using high-order time discretization and random-data formulations.

The voltage-based Neural Field Equation (NFE) is a continuum model of large-scale neural tissue in which the state variable is the membrane-potential, or voltage, field over space and time. In its standard form, the dynamics combine a local leak term with a spatially nonlocal synaptic term, and the framework admits extensions with external input, finite-speed propagation, dendritic cable processing, extracellular electrodynamics, and random data. It also appears as a continuum limit of microscopic leaky integrate-and-fire and Hawkes-process descriptions, so the voltage-based NFE occupies a position between cellular biophysics, population dynamics, and observation models for extracellular potentials [(Graben et al., 2013); (Luçon et al., 25 Jul 2025)].

1. Canonical formulations

A standard voltage-based NFE on the real line is

$$\partial_t u(t,x) =-\,u(t,x)+\int_{\mathbb R}W(x-y)\,f\bigl(u(t,y)\bigr)\,dy,$$

where $u=u(t,x)$ is the membrane-potential field, $W\colon\mathbb R\to[0,\infty)$ is a symmetric connectivity kernel, and $f\colon\mathbb R\to[0,1]$ is a smooth sigmoid (Luçon et al., 25 Jul 2025). In this formulation the state variable is voltage itself, and the nonlinearity is applied before spatial averaging.

A two-dimensional formulation on a bounded domain $\Omega\subset\mathbb R^2$ writes

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$

and, with finite-speed propagation,

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$

with history condition $V(x,t)=V_0(x,t)$ for $t\in[-\tau_{\max},0]$ and space-dependent delay $\tau(x,y)=\|x-y\|_2/v$ (Lima et al., 2015). Here $u=u(t,x)$0 is the membrane time-constant, $u=u(t,x)$1 is external input, $u=u(t,x)$2 is a homogeneous synaptic kernel depending only on distance, and $u=u(t,x)$3 is a smooth or piecewise smooth firing-rate function.

A random-data formulation treats the same voltage-based equation pathwise on a probability space:

$u=u(t,x)$4

with randomness in the synaptic kernel, firing-rate function, external stimulus, and initial voltage (Avitabile et al., 22 May 2025). The associated abstract Cauchy problem is posed on $u=u(t,x)$5.

These formulations share the same structural motif: a local relaxation term and a spatially aggregated recurrent input. A common misconception is that voltage-based and activity-based neural fields are interchangeable. The data support a narrower statement: they are distinct equations, although in the traveling-wave setting there is a one-to-one correspondence between waves of the voltage-based NFE and those of the rate-based NFE (Luçon et al., 25 Jul 2025).

2. Microscopic derivation and continuum limit

A biophysically explicit derivation starts from a three-compartment pyramidal neuron with apical dendrite, somatic/basal dendrite, and axon hillock, with membrane potentials $u=u(t,x)$6 and mesh currents $u=u(t,x)$7 satisfying Kirchhoff’s laws (Graben et al., 2013). Excitatory and inhibitory postsynaptic currents are written as

$u=u(t,x)$8

with $u=u(t,x)$9 and $W\colon\mathbb R\to[0,\infty)$0 given by synaptic impulse responses convolved with presynaptic spike trains.

Under the simplifications of neglecting synaptic shunting in the transition-matrix, approximating $W\colon\mathbb R\to[0,\infty)$1, collapsing excitatory and inhibitory kernels into $W\colon\mathbb R\to[0,\infty)$2, absorbing constants into weights $W\colon\mathbb R\to[0,\infty)$3, and setting diffusion sources $W\colon\mathbb R\to[0,\infty)$4, the network reduces to leaky integrate-and-fire dynamics:

$W\colon\mathbb R\to[0,\infty)$5

The same derivation yields a dendritic field potential $W\colon\mathbb R\to[0,\infty)$6 as a filtered combination of spike trains and membrane voltage.

Passing to the continuum limit over populations distributed on cortical sheets $W\colon\mathbb R\to[0,\infty)$7, discrete variables become fields $W\colon\mathbb R\to[0,\infty)$8, synaptic weights become kernels $W\colon\mathbb R\to[0,\infty)$9, and spike output is replaced by a smooth activation function

$f\colon\mathbb R\to[0,1]$0

The resulting Amari-type neural-field equation is

$f\colon\mathbb R\to[0,1]$1

This derivation is significant because it shows that the voltage-based NFE need not be introduced only as a phenomenological nonlocal evolution equation. In this setting it is the continuum image of a specific compartmental circuit model with identifiable electrotonic parameters, synaptic kernels, and observation variables.

3. Dendritic-processing formulations

A major generalization replaces a purely somatic state variable by a dendritic voltage field $f\colon\mathbb R\to[0,1]$2, where $f\colon\mathbb R\to[0,1]$3 is the somatic coordinate and $f\colon\mathbb R\to[0,1]$4 is depth along an unbranched dendritic cable. The governing integro-differential model is

$f\colon\mathbb R\to[0,1]$5

with leak rate $f\colon\mathbb R\to[0,1]$6, electrotonic diffusivity $f\colon\mathbb R\to[0,1]$7, external input $f\colon\mathbb R\to[0,1]$8, and homogeneous Neumann boundary conditions in $f\colon\mathbb R\to[0,1]$9 in the numerical setting (Avitabile et al., 2020). For axo-dendritic interactions concentrated at somatic and contact layers,

$\Omega\subset\mathbb R^2$0

so the nonlocal operator reduces to a somatic convolution delivered back to dendritic depth $\Omega\subset\mathbb R^2$1.

A related analytical formulation is posed on $\Omega\subset\mathbb R^2$2, $\Omega\subset\mathbb R^2$3, with

$\Omega\subset\mathbb R^2$4

where

$\Omega\subset\mathbb R^2$5

and $\Omega\subset\mathbb R^2$6, $\Omega\subset\mathbb R^2$7 smooth, bounded, and Lipschitz (Avitabile et al., 2024). The weak formulation uses

$\Omega\subset\mathbb R^2$8

and a weak solution satisfies

$\Omega\subset\mathbb R^2$9

For the singular problem $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$0, there is a unique solution

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$1

while for the regular problem $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$2 there is a unique weak solution in $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$3 with the stated energy bound (Avitabile et al., 2024). The same analysis proves continuous dependence

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$4

and, when $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$5,

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$6

These models alter the geometry of the voltage-based NFE in a substantive way. The diffusion operator acts only in the dendritic direction, and the synaptic term remains nonlocal, so the problem is markedly different from classical neural field equations. Numerical experiments report that for $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$7 a two-bump profile collapses to a single peak at $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$8, while small $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t))\,dy,$$9 merges the bumps and broadens the profile; the error $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$0 scales linearly in $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$1, confirming the $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$2 estimate in norm (Avitabile et al., 2024).

4. Electrodynamics and extracellular observation

In an electrodynamic extension, the voltage-based NFE is coupled to extracellular charge transport and field potentials. The driving dipole-current density in cortical tissue is represented by a current field $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$3 derived from dendritic trunk currents, while the extracellular medium supports both diffusive and Ohmic currents:

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$4

where $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$5 is charge density, $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$6 is the diffusion coefficient, $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$7 is electrical conductivity, and $$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$8 is extracellular potential (Graben et al., 2013). The total current density obeys the Nernst–Planck relation

$$c\,\partial_t V(x,t)=I(x,t)-V(x,t)+\int_\Omega K(\|x-y\|_2)\,S(V(y,t-\tau(x,y)))\,dy,$$9

Charge conservation gives

$V(x,t)=V_0(x,t)$0

and Gauss’s law in a linear dielectric is

$V(x,t)=V_0(x,t)$1

These equations couple the extracellular potential $V(x,t)=V_0(x,t)$2, the charge density $V(x,t)=V_0(x,t)$3, and the neural-field driving current.

The final voltage-neural-field equation with electrodynamic feedback augments classical nonlocal coupling by a diffusion-induced term evaluated in cortical layer 1:

$V(x,t)=V_0(x,t)$4

The extracellular potential satisfies

$V(x,t)=V_0(x,t)$5

The local field potential at an electrode position $V(x,t)=V_0(x,t)$6 is then an observation variable,

$V(x,t)=V_0(x,t)$7

obtained from the Green’s function of $V(x,t)=V_0(x,t)$8. In a homogeneous resistive medium,

$V(x,t)=V_0(x,t)$9

so the LFP is determined by the divergence of the neural current density. This directly addresses the frequent simplification that extracellular potentials can be read off from membrane voltage alone; in this framework they require a separate observation model based on dipole currents, charge density, and extracellular transport.

5. Traveling waves, stability, and Hawkes limits

For the standard voltage-based NFE on $t\in[-\tau_{\max},0]$0, traveling-wave solutions take the form

$t\in[-\tau_{\max},0]$1

where the profile satisfies

$t\in[-\tau_{\max},0]$2

with $t\in[-\tau_{\max},0]$3 (Luçon et al., 25 Jul 2025). Under the stated hypotheses on $t\in[-\tau_{\max},0]$4 and on a bistable sigmoid $t\in[-\tau_{\max},0]$5, there is a unique solution modulo translation, and the speed is

$t\in[-\tau_{\max},0]$6

In the neutral case, the symmetry condition

$t\in[-\tau_{\max},0]$7

implies $t\in[-\tau_{\max},0]$8, so the traveling wave reduces to a family of stationary translated profiles

$t\in[-\tau_{\max},0]$9

The linearized operator

$\tau(x,y)=\|x-y\|_2/v$0

has a spectral gap in the weighted space $\tau(x,y)=\|x-y\|_2/v$1, $\tau(x,y)=\|x-y\|_2/v$2, yielding contraction on the orthogonal complement of $\tau(x,y)=\|x-y\|_2/v$3. This underlies the normally hyperbolic manifold structure of the translation family, the existence of a smooth isochron map $\tau(x,y)=\|x-y\|_2/v$4, and local exponential attraction to the manifold.

A microscopic realization uses interacting Hawkes processes on a lattice $\tau(x,y)=\|x-y\|_2/v$5, with spike intensity

$\tau(x,y)=\|x-y\|_2/v$6

and voltage

$\tau(x,y)=\|x-y\|_2/v$7

On any fixed time window, the piecewise-constant profile $\tau(x,y)=\|x-y\|_2/v$8 converges in probability in $\tau(x,y)=\|x-y\|_2/v$9 to the solution of the voltage-based NFE. On the diffusive time scale $u=u(t,x)$00 in the neutral case, the phase performs Brownian wandering: there exists a càdlàg process $u=u(t,x)$01 converging in law to a standard Brownian motion, and the profile remains close to a translated deterministic wave with diffusion coefficient

$u=u(t,x)$02

The same source states that there is a one-to-one correspondence between traveling waves of the voltage-based NFE and those of the rate-based NFE

$u=u(t,x)$03

with the same speed $u=u(t,x)$04. This correspondence is specific and does not collapse the distinction between voltage-based and rate-based formulations.

6. Numerical analysis and random-data frameworks

For the two-dimensional delayed voltage-based NFE, a high-order implicit-in-time discretization uses the three-level backward-difference formula

$u=u(t,x)$05

with the right-hand side evaluated at the new time level (Lima et al., 2015). Writing

$u=u(t,x)$06

the non-delay scheme becomes

$u=u(t,x)$07

or equivalently

$u=u(t,x)$08

The fixed-point iteration

$u=u(t,x)$09

converges linearly under the step-size restriction

$u=u(t,x)$10

For non-integer delays, $u=u(t,x)$11 and the delayed argument is approximated by linear interpolation, maintaining $u=u(t,x)$12 local accuracy in time.

The spatial discretization partitions $u=u(t,x)$13 into uniform cells with tensor-product Gauss-Legendre quadrature. A direct Nyström discretization would cost $u=u(t,x)$14 operations per iteration, so the method applies Chebyshev interpolation on a smaller set of $u=u(t,x)$15 nodes, reducing each fixed-point iteration to

$u=u(t,x)$16

when $u=u(t,x)$17 (Lima et al., 2015). Under the stated smoothness and contractivity conditions,

$u=u(t,x)$18

The reported numerical tests confirm second-order time convergence, $u=u(t,x)$19 spatial behavior for $u=u(t,x)$20, and stable delayed simulations with 2–4 inner fixed-point iterations.

A complementary framework treats random-data voltage-based NFEs as abstract Cauchy problems on $u=u(t,x)$21, with strongly measurable random inputs $u=u(t,x)$22 that are assumed mutually independent (Avitabile et al., 22 May 2025). Under Hypotheses III, there exists a unique strongly measurable solution

$u=u(t,x)$23

satisfying pathwise bounds

$u=u(t,x)$24

If the inputs belong to the corresponding $u=u(t,x)$25 Bochner spaces, then

$u=u(t,x)$26

The same analysis extends to semi-discrete approximations on finite-dimensional subspaces $u=u(t,x)$27, yielding unique strongly measurable solutions $u=u(t,x)$28 with uniform $u=u(t,x)$29 bounds when $u=u(t,x)$30. Under finite-dimensional noise, the solution takes parametric form $u=u(t,x)$31, providing a foundation for Monte Carlo sampling, stochastic collocation, and stochastic Galerkin schemes (Avitabile et al., 22 May 2025).

7. Finite-voltage mean-field extensions

A recent exact mean-field construction introduces a two-phase quadratic integrate-and-fire neuron whose membrane potential evolves according to two alternating Riccati equations within finite bounds, yet still admits a low-dimensional description governed by a single complex Riccati equation,

$u=u(t,x)$32

with collective quantities such as firing rate $u=u(t,x)$33 and mean voltage $u=u(t,x)$34 recovered as explicit functions of $u=u(t,x)$35 (Cestnik, 4 Mar 2026). In the commonly studied example,

$u=u(t,x)$36

so that

$u=u(t,x)$37

In a proposed interpretation as a voltage-based neural field, one can promote $u=u(t,x)$38, replace chemical coupling by

$u=u(t,x)$39

add electrical coupling through $u=u(t,x)$40, and allow space-dependent drives. The resulting neural-field Riccati equation is

$u=u(t,x)$41

with $u=u(t,x)$42 and $u=u(t,x)$43 given pointwise by the same algebraic relations (Cestnik, 4 Mar 2026).

This suggests a direction in which voltage-based neural fields can inherit exact finite-dimensional closures from mean-field spiking models while retaining nonlocal coupling and voltage diffusion. In the formulation stated in the source, chemical synapses enter additively into $u=u(t,x)$44, while electrical coupling modifies $u=u(t,x)$45 or produces a diffusion of $u=u(t,x)$46.