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Excitability across scales: a unifying EI model

Thomas S. J. Burger, Rodolphe Sepulchre

arXiv:2610.13275v1q-bio.NCeess.SY

TL;DR

Different principles for modeling cellular and population excitability make it difficult to study how microscopic mechanisms shape macroscopic activity. The paper introduces a shared conductance-based EI framework using scale-dependent memductances. It reports that neuronal excitability and synaptic interconnections jointly contribute to population events with their own scales, robust to substantial parameter heterogeneity.

  • Problem

    Different modeling principles for cellular and population excitability make it difficult to study how cellular mechanisms shape population behavior without simulating entire populations at the neuronal scale.

    Source: §1 · p. 2

  • Method

    The EI model unifies cellular and population excitability through a Hodgkin–Huxley-style circuit architecture with scale-dependent temporal and spatial memductances.

    Source: §1 · p. 2, Conclusion · p. 19

  • Results

    Neuronal excitability and synaptic interconnections both contribute to population spikes with distinct temporal and spatial scales that remain robust under broad intrinsic and synaptic conductance heterogeneity.

    Source: Conclusion · p. 19

  • Takeaways & Limitations

    Separating circuit architecture from modeling scale provides a physically interpretable framework for analyzing and designing temporal, spatial, and population excitability.

    Source: Conclusion · pp. 19–20

    The model remains to be connected and compared quantitatively with electrophysiological and imaging data, reduced spiking models, neural-field models, and detailed conductance-based descriptions.

    Source: Conclusion · pp. 19–20

Abstract

from arXiv · show

Neuronal excitability is commonly modeled using different principles at the cellular and population scales, making it difficult to study how microscopic mechanisms shape macroscopic activity. We introduce a spatiotemporal excitatory-inhibitory (EI) model that unifies these scales through a common mixed-feedback architecture. The model retains the conductance-based circuit structure of Hodgkin-Huxley models and describes intrinsic and synaptic currents through memductances whose temporal and spatial voltage dependence encodes dynamics at the relevant scale. In its minimal form, temporal excitability arises from fast excitatory feedback opposed by slower inhibitory feedback, while spatial excitability arises from short-range excitation opposed by longer-range inhibition, recovering the core lateral-inhibition motif of Amari neural fields. We show that combining these mechanisms gives rise to population excitability in strongly heterogeneous populations. In particular, the model generates population spikes that are localized in both space and time and possess population-level scales distinct from those of individual neuronal spikes, despite substantial cell-to-cell parameter variability. We then show that both intrinsic temporal excitability and spatial synaptic feedback are required for coherent mean-field events and robust spatiotemporal population rhythms. Finally, selective modulation of memductances in a subset of the population tunes the frequency of the population rhythm, illustrating how changes at the cellular scale can shape macroscopic dynamics. By separating circuit architecture from modeling scale, the proposed framework provides a physically interpretable approach to analyzing and designing neuronal excitability across scales.

1 Introduction

Cellular and population excitability are usually modeled using different principles, complicating analysis of how cellular mechanisms shape population behavior. The paper proposes a shared, biophysically interpretable circuit framework whose temporal and spatial memductances can be represented at different scales.

1 Introduction

  • Different modeling principles at cellular and population scales make it difficult to study how cellular excitability mechanisms shape population excitability.Simulating entire populations at the neuronal scale to study cross-scale regulation is cumbersome and not scalable.Source: §1 · p. 2
  • The proposed spatiotemporal model unifies the temporal Hodgkin–Huxley model and the spatial Amari model.Source: §1 · p. 2
  • The model retains Hodgkin–Huxley’s capacitive circuit with voltage-gated current sources, whose history-dependent conductances the paper calls memductances.It treats ion channels as internal memductances and synapses as external memductances with temporal and spatial voltage dependence.Source: §1 · pp. 2–3
  • The framework places temporal and spatial complexity in memductance voltage dependence, allowing simpler scale-specific models to preserve the same physical circuit structure.Temporal models spatially average the dependence, while spatial models temporally average it; the authors argue that resolution can be adjusted through the memductances.Source: §1 · p. 3
  • The paper uses the separable memductance design to investigate which neuronal properties support robust mean-field phenomena and how cellular properties relate to population dynamics.Source: §1 · p. 3

2 The mixed-feedback motif of temporal excitability

The temporal mixed-feedback motif combines fast excitation with slower inhibition to produce complementary excitatory and rebound thresholds. A simplified EI model preserves this behavior under broad parameter conditions while retaining an interpretable conductance-based circuit structure.

2 The mixed-feedback motif of temporal excitability

  • A capacitive membrane in feedback with oppositely signed sodium and potassium currents forms the Hodgkin–Huxley mixed-feedback circuit.Sodium activation supplies positive feedback for the spike’s upward stroke; potassium activation provides negative feedback for repolarization.Source: §2 · pp. 3–4
  • The EI model exhibits complementary excitatory and rebound thresholds, and rebound excitability is lost without inhibitory current.The minimal model retains the threshold properties illustrated for the Hodgkin–Huxley model.Source: §2 · p. 5, Figure 3 · p. 6
  • The simplified EI model replaces sodium and potassium conductances with excitatory and inhibitory memductances, with excitation activating faster than inhibition.Each memductance is modeled as a first-order lag followed by a sigmoid; the minimal model omits sodium inactivation.Source: §2 · p. 5
  • Both thresholds require sufficient timescale separation and overlapping activation functions that invert the excitatory–inhibitory balance near equilibrium.The activation curves support inhibition reduction by hyperpolarization and rapid excitation during rebound or depolarization.Source: §2 · p. 5
  • The EI model achieves both excitability mechanisms without separate inactivation, deinactivation, or hyperpolarization-activated variables.Its stated robustness comes from broad voltage ranges for reducing inhibition and entering steep excitation, while preserving fast positive and slower negative feedback.Source: §2 · p. 7

3 The mixed-feedback motif of lateral inhibition

Amari’s neural-field model expresses spatial excitability through lateral inhibition: short-range excitation and long-range inhibition create a spatial threshold event. The interaction kernel determines the width of active regions while suppressing activity elsewhere.

3 The mixed-feedback motif of lateral inhibition

  • Spatially structured excitation and inhibition can produce activity patterns such as oscillations, with lateral inhibition as a foundational circuit motif.The section situates this approach alongside the Wilson–Cowan and Amari models.Source: §3 · p. 7
  • Amari’s spatial model uses a firing-rate function and spatial convolution with an interaction kernel.Source: §3 · p. 7
  • Short-range excitation opposed by long-range inhibition creates a threshold event localized in space, with kernel-dependent active-region width and suppression elsewhere.Source: §3 · p. 7

4 A mixed-feedback motif of spatiotemporal excitability

The model combines pointwise temporal mixed feedback with spatially filtered excitatory–inhibitory coupling. It concentrates scale-dependent dynamics in memductances while retaining a common circuit architecture.

4 A mixed-feedback motif of spatiotemporal excitability

  • The mixed-feedback motif uses separate excitatory and inhibitory populations with a spatial kernel, and the kernel modulates memductance rather than current.The synaptic memductances are described as modulating currents that model interneuron coupling within the population.Source: §4 · p. 8
  • Lateral inhibition requires a short spatial excitatory scale relative to inhibition, while temporal excitability requires a short excitatory timescale relative to inhibition.Combining these spatial and temporal conditions yields the spatiotemporal mixed-feedback model.Source: §4 · p. 8
  • Internal memductances depend on voltage history at each position, whereas synaptic memductances depend on voltage history in a spatial neighborhood.The model denotes spatial memductances with uppercase E/I and purely temporal internal memductances with lowercase e/i.Source: §4 · pp. 8–9
  • The canonical architecture combines local temporal mixed feedback with spatially filtered excitatory–inhibitory coupling to determine network dynamics.Pointwise internal currents shape local excitable dynamics, while synaptic currents recruit and suppress activity across neighboring positions.Source: §4 · pp. 8–9, Figure 6 · p. 9
  • The model uses a time convolution and nonlinearity for internal currents, and time and space convolutions followed by a nonlinearity for synaptic currents.This structure is described as straightforward to analyze and computationally inexpensive to simulate.Source: §4 · p. 9

5 Population Excitability

The heterogeneous EI population forms localized space–time population spikes, while ablations show that coherent mean-field activity requires both temporal and spatial memductances. Rescaling a subset of neurons’ intrinsic conductances then tunes the population-rhythm frequency.

5.1 Population spike

  • Despite ±40% variability in internal memductances, the heterogeneous population forms a coherent event bounded in space and time.A localized group is recruited and returns toward rest after a finite interval.Source: §5.1 · p. 10, Figure 7 · p. 11
  • Tpop = 3.3 ms and ℓpop = 0.46, compared with Tind = 1.3 ms; the population spike has distinct spatial and temporal scales.Widths are measured at full width at half maximum, with Tind the median across uncoupled neurons.Source: §5.1 · p. 12

5.2 Complementarity of temporal and spatial mixed-feedback

  • The ablations test whether the collective event reflects mixed-feedback architecture rather than coincident responses to a shared input.The comparison removes spatial or temporal excitability in turn.Source: §5.2 · p. 12

5.2.1 Complementarity of internal and synaptic currents

  • Without spatial memductances, independently responding units lose coherent spatial activity despite retaining local internal memductances.Any apparent spatial agreement is accidental and follows their responses to the same input.Source: §5.2.1 · p. 12

5.2.2 Complementarity of internal and interneuronal feedback

  • Without temporal memductances, a long input produces only a brief surge before spike times desynchronize and trial-averaged voltage returns near rest.The comparison averages responses over 100 trials.Source: §5.2.2 · p. 14, Figure 10 · p. 16

5.2.3 No mean-field excitability without both internal and synaptic currents

  • Both temporal and spatial memductances are required for mean-field events; removing either leaves the coherence measure without a transient peak.With both present, the heterogeneous network produces a sequence of population spikes whose amplitude gradually decays.Source: Figure 11 · p. 17, §5.2.3 · p. 17

5.3 Tuning through heterogeneity

  • Rescaling intrinsic conductances in a population fraction decreases rhythm frequency, with the shift depending on the scaling factor and affected fraction.Robustness to heterogeneity allows individual neurons to act as an accelerator or brake on the population response.Source: §5.3 · p. 18

Conclusion

The EI model separates circuit architecture from the scale of excitability by combining cellular feedback with spatial synaptic interactions. Its population events have distinct spatial and temporal scales, remain robust under heterogeneity, and motivate further empirical and quantitative comparisons.

Conclusion

  • The EI model separates circuit architecture from scale, using a Hodgkin–Huxley-like capacitive membrane and opposing memristive currents with scale-specific memductance dynamics.It provides a framework for analyzing and designing temporal excitability, spatial excitability, and their interaction.Source: Conclusion · p. 19
  • Fast positive feedback tempered by slower negative feedback produces temporal excitability, while short-range excitation opposed by longer-range inhibition produces spatial excitability.Together, these mechanisms give rise to population excitability as spatially nearby neurons produce temporally synchronized events.Source: Conclusion · p. 19
  • Population mean-field spikes have spatial and temporal scales distinct from individual neuronal spikes and remain robust despite broad heterogeneity in intrinsic and synaptic conductances.The model’s population heterogeneity is proposed as a tunable knob for mean-field dynamics.Source: Conclusion · p. 19
  • The EI model is presented as a candidate bridge between conductance-based cellular neurophysiology and population-scale mean-field models.The framework combines a physically interpretable conductance-based architecture with explicit temporal and spatial memductance dependence.Source: Conclusion · pp. 19–20
  • The authors propose connecting the model to electrophysiological and imaging data to test whether its feedback structures occur in biological tissue and predict event robustness or tunability.They also identify quantitative comparison with reduced spiking, classical neural-field, and detailed conductance-based models as future work.Source: Conclusion · pp. 19–20

A.1 Hodgkin–Huxley model

The Hodgkin–Huxley appendix specifies standard channel kinetics, parameter sources, and simulation conditions. It also describes the smooth pulse inputs used for Figure 2.

A.1 Hodgkin–Huxley model

  • The sodium activation and inactivation variables m and h, and potassium activation variable n, follow standard Hodgkin–Huxley kinetics.The voltage-dependent transition rates are listed in equations A.2–A.4.Source: §A.1 · p. 20
  • The Hodgkin–Huxley simulation parameters are given in Table 1, with capacitance in µF cm−2, maximal conductances in mS cm−2, and reversal potentials in mV.Table 1 is identified as the parameter table for the Hodgkin–Huxley neuron model.Source: Table 1 · p. 20, §A.1 · p. 20
  • Figure 2’s applied current is constructed from smooth rectangular pulses, with pulse amplitudes measured in µA cm−2 and times in ms.The pulse construction uses a logistic function with k = 10 ms−1.Source: §A.1 · p. 21
  • The model starts at v(0) = −65 mV, m(0) = 0.05, h(0) = 0.6, and n(0) = 0.32, and is integrated for 100 ms with Tsit5 at 10−6 tolerances.The solution is sampled every 0.1 ms.Source: §A.1 · p. 21

A.2 Amari’s model

The Amari neural-field appendix specifies its firing-rate function, spatial interaction kernel, and localized inputs. It also records the domain and numerical setup used for Figure 5.

A.2 Amari’s model

  • The firing-rate function for the Amari neural-field simulations in Figure 5 is a smooth approximation of a Heaviside threshold.Source: §A.2 · p. 21
  • Spatial coupling uses a Mexican-hat kernel with A = 2, B = 1, a = 0.3, b = 2, and field time constant τ = 3.Source: §A.2 · p. 21
  • The external input consists of two spatially localized Gaussian perturbations with amplitudes (2.5, 2.6), times (10, 210), and σx = σt = 5.Source: §A.2 · p. 21
  • The field spans x ∈ [−20, 20] at 501 points with periodic boundaries and circular-convolution coupling evaluated by FFT.It starts at v(x, 0) = 0 and is integrated to t = 400 with Tsit5 at 10−6 tolerances, sampled every 0.5.Source: §A.2 · p. 21

A.3 EI model

The EI-model appendix points to the tables containing its intrinsic conductance and temporal simulation parameters. It also states the default integration method and tolerances.

A.3 EI model

  • The EI model’s intrinsic conductance parameters are listed in Table 2.Source: §A.3 · p. 21
  • Table 2 is identified as the source of temporal parameters used in the EI-model simulations.Source: Table 2 · p. 21
  • Unless stated otherwise, EI-model simulations use Tsit5 with relative and absolute tolerances of 10−5.Source: §A.3 · p. 21

A.3.1 EI model excitability criterion

The EI model supports excitation when fast excitatory feedback produces negative differential conductance, while slower inhibition initially lags. Rebound excitation additionally requires inhibition to be sufficiently deactivated, followed by inhibitory recovery that restores positive differential conductance.

A.3.1 EI model excitability criterion

  • Excitatory feedback can amplify a rising voltage when its reversal potential exceeds that voltage and its steep, strongly coupled activation overcomes leak.The excitatory sigmoid must be sufficiently steep and sufficiently strongly coupled to overcome the leak.Source: §A.3.1 · p. 22
  • Recovery requires slow inhibition to catch up during depolarization, restoring positive differential conductance.Because Ei < 0, the inhibitory contribution is positive at depolarized voltages.Source: §A.3.1 · pp. 22–23
  • During early rebound, slow inhibition remains near its hyperpolarized value while excitation tracks voltage; rebound amplification requires negative differential conductance in that phase.The hyperpolarizing pulse must deactivate slow inhibition sufficiently for the fast excitatory current to create negative differential conductance.Source: §A.3.1 · p. 23
  • Rebound excitation followed by recovery requires effective differential conductance to change from negative during early rebound to positive later.The early-rebound and recovery values are evaluated at vrise and vrec, respectively.Source: §A.3.1 · p. 24

A.3.2 Excitability simulations

The simulations test spatiotemporal excitability under heterogeneous parameters and compare complete and simplified model variants. Additional experiments measure coherent population activity and rhythm frequency under selective conductance rescaling.

A.3.2 Excitability simulations

  • The spatiotemporal simulations used a 901-point spatial grid and introduced independent ±40% variation in intrinsic conductances while keeping maximal synaptic conductances homogeneous.The excitatory field received the applied input, while the inhibitory field received no direct applied current.Source: §A.3.2 · pp. 24–25
  • For Figures 8 and 9, the simulations respectively removed spatial interactions or replaced dynamic intrinsic conductances with an effective leak while retaining synaptic coupling.Figure 9 used an effective reversal potential chosen to preserve total resting intrinsic current.Source: §A.3.2 · p. 25
  • In Figure 10, 100 paired trials varied intrinsic conductances and baseline synaptic gains independently by ±40%, comparing the canonical and purely synaptic models under matched parameters and input.The plotted measure was excitatory voltage at x0 = 0, averaged pointwise across trials.Source: §A.3.2 · pp. 25–26
  • Figure 11 compares the canonical model with temporal-only and spatial-only variants under the same heterogeneous realization to quantify coherent population activity.The temporal-only condition removes spatial synaptic conductances; the spatial-only condition replaces dynamic intrinsic conductances with matched spatial synaptic channels.Source: §A.3.2 · p. 26
  • Figure 12 varies conductance scaling across 20 values and central domain widths of 10%, 20%, or 30%, then estimates rhythm frequency from coherent spatial power.The initial 30% of each simulation was discarded, and rhythms were retained only with at least three intervals and coefficient of variation no greater than 0.5; failed conditions were assigned frequency zero.Source: §A.3.2 · p. 26
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