Title : The Bansal neuron model: Nonlinear membrane charge, field-to-neuron coupling, and multi-resonant dynamics
Abstract:
Neuronal excitability depends on the interaction of membrane charge storage, voltage-dependent ionic transport, internal electrical relaxation, and the response to applied stimulation. Although these mechanisms can influence the same voltage trace, a change in firing pattern does not by itself identify which mechanism produced it. In this presentation, I introduce the Bansal Neuron Model as a specified field-to-neuron construction designed to distinguish these contributions through governing equations, matched controls, and reproducible numerical tests.
The starting point is a nonlinear membrane charge law obtained from a declared dielectric boundary-value problem. Terminal charge is recovered from the field solution and differentiated to obtain positive differential capacitance. This constitutive relation is coupled to conventional Hodgkin–Huxley sodium, potassium, and leakage currents, with optional passive relaxation and up to two DC-blocked resonant branches. The resulting point-neuron family has an exact matched Hodgkin–Huxley restriction, while its fully enabled realization contains nine dynamical states. I explain how the F1–F5 foundational architectures separates representation and recovery, governing laws, parameter inference, numerical field construction, and neuronal response, without treating these roles as independently validated new biological mechanisms.
A central feature is the controlled separation of nonlinear storage from delayed electrical response. Twelve matched configurations vary constant versus field-derived charge storage, relaxation off versus on, and zero, one, or two resonant branches, while preserving the specified ionic kinetics and stimulation conditions. Conventional first- and second-order finite-element calculations independently examine approximation of the dielectric field, terminal charge, and the effect of numerical error on neuronal observables. The presentation connects this field calculation directly to computed membrane voltage, spike timing, impedance, and local stability.
Under a 200-millisecond current step of 10 microamperes per square centimetre, the matched Hodgkin–Huxley reference and the full nine-state realization each produce fourteen accepted spikes. Their first-spike latencies, however, differ: approximately 1.901 and 2.121 milliseconds, respectively. Local equilibrium continuation places the corresponding Hopf crossings near 9.775 and 11.111 microamperes per square centimetre. The nearby cycles computed for the complete realization exhibit an unstable transverse mode; the shift in crossing location must therefore not be interpreted as a demonstrated stable oscillatory regime. Across nineteen operational response protocols, the model family yields nine positive signatures, nine negative outcomes in the searches performed, and one indeterminate outcome. Eight positive criteria are also satisfied by the Hodgkin–Huxley restriction, while selected resonant configurations additionally satisfy an isolated rebound-doublet criterion.
Throughout the talk, I distinguish changes in simulated observables from improvements in predictive accuracy and use mechanism ablations to expose each enabled branch's contribution.
These findings illustrate what can be established by mechanism-resolved computational comparisons and what remains open. The numerical results support a specified coupling of field-derived charge, neuronal dynamics, relaxation, and resonant states; they do not establish biological identity for the synthetic dielectric or resonant mechanisms, superiority on independent electrophysiological recordings, or universal recovery of existing neuron models. I conclude by outlining experimentally testable paths involving voltage-clamp charge measurements, frequency-dependent impedance, and held-out current-clamp responses, with particular attention to parameter identifiability, numerical convergence, and fair comparison against established models.
Keywords: Bansal Neuron Model; Neuronal Electrophysiology; Nonlinear Membrane Charge; Differential Capacitance; Hodgkin–Huxley Model; Multiphysics Modelling; Multi-Resonant Dynamics; Hopf Bifurcation; Computational Neuroscience.
