A memristive model of spatio-temporal excitability

Amir Shahhosseini; Rodolphe Sepulchre; Thomas SJ Burger

REVIEW 4 major objections 5 minor 15 references

A memristive model of spatio-temporal excitability

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The same memristive mixed-feedback circuit that produces temporal spiking also produces spatial bumps in a neural field.

desk verdict A genuinely new unified memristive framework for temporal and spatial excitability, with a solid temporal analysis but an under-validated spatial branch that needs either an equivalence argument or systematic comparison to Amari. read the letter →

arxiv 2505.22269 v1 pith:DO7OZZNG submitted 2025-05-28 eess.SY cs.SY

Thomas SJ Burger , Amir Shahhosseini , Rodolphe Sepulchre This is my paper · ORCID

classification eess.SYcs.SY MSC 92C2037N25

keywords memristiveexcitabilitymemductancemixedfeedbackspatio-temporalmodelneuralfieldconvolutionnetworktwo-population

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes that neuronal excitability — the all-or-nothing pulse response to a threshold stimulus — is not two separate phenomena at the cellular and population scales but one mechanism appearing in two guises. It builds a single memristive circuit model whose currents obey Ohm's law with conductances that carry memory ('memductances'), and shows that tuning only those memductances reproduces the temporal excitability of the classic 1952 biophysical neuron model and the spatial excitability of a lateral-inhibition neural-field model. A small stimulus below threshold gives a small response; a slightly larger stimulus triggers a stereotyped pulse in time or a localized persistent bump in space. If the model is right, the same mixed-feedback architecture — fast positive feedback followed by slower negative feedback — can be used to simulate, analyze, and design excitable behaviors across scales, from ion channels to populations.

What carries the argument

The central object is the memductance: a conductance that obeys Ohm's law (current equals g times (v − E)) but whose value depends on the history of the entire neural field, so it carries memory. The paper models each memductance as a feedforward nonlinear convolution operator — a small CNN. The architecture is a nonlinear RC circuit with two such memductances, e and i: e is fast, inward, and positive-feedback; i is slow, outward, and negative-feedback. For spatial excitability, the e and i memductances are driven by two separate populations and convolved with exponential kernels of short and long range. All modeling complexity is concentrated in the memductances; the circuit skeleton stays fixed, which is what gives the model its cross-scale reach.

What would settle it

Simulate the two-population spatial memristive model and the original lateral-inhibition neural-field model side by side over a grid of input amplitudes, kernel widths, and thresholds; if there is any parameter region where the original model produces a localized persistent bump but the memristive model does not (or vice versa), the claimed spatial unification fails. A single matching example, as in the paper, is not enough.

Watch

Extended reading notes

Core claim

The central discovery claim is that temporal and spatial excitability share a single mechanism: a mixed-feedback loop where fast positive feedback provides the switch and slower negative feedback provides recovery and containment. In time, the positive feedback is a fast inward current and the negative feedback is a slower outward current; in space, the positive feedback is short-range excitation and the negative feedback is long-range inhibition. The paper's unified model encodes both in one RC circuit with two memristive currents, labeled e and i, whose memductances are nonlinear convolution operators (simple CNN models). A purely temporal instantiation of the memductances yields the canonical firing threshold, a spatial instantiation on separate excitatory and inhibitory populations yields the lateral-inhibition bump, and combining them yields both. The claim is that the mechanism of excitability is the mixed-feedback architecture itself, and that this mechanism is robust to the details of the memductance model.

Load-bearing premise

The claim rests on assuming that splitting Amari's signed spatial interaction kernel into two separate nonnegative excitatory and inhibitory memductance kernels reproduces the same spatial excitability; the paper shows one simulation with handpicked parameters rather than a proof or systematic comparison.

Editorial extensions

If this is right

The same circuit model can reproduce both the temporal firing threshold of a biophysical neuron and the spatial bump of a lateral-inhibition neural field, by changing only the memductance maps.
Because the fixed RC circuit is preserved across scales, temporal and spatial models become special cases of one architecture rather than two different modeling traditions.
Excitability is robust to the details of the memductance model, so simplified memductance representations can be used for large-scale simulation without losing the fundamental behavior.
The spatio-temporal model gives a single framework for studying how cellular-scale ion-channel properties shape population-scale activity.
Temporal memductances map to ion channels and spatial memductances map to synaptic currents, preserving biophysical interpretability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If this unification holds, excitability may be a design principle rather than a material property: any system with fast positive and slow negative feedback could show the same threshold behavior, whether in neurons, populations, or engineered circuits.
A natural testable extension is to use the CNN memductance representation to model traveling waves and oscillations in the neural field, which the paper mentions but does not demonstrate for the unified model.
The two-population nonnegative memductance construction might also be applied to other signed-kernel neural-field models, potentially replacing signed interaction kernels with physiologically interpretable excitatory and inhibitory populations.
The operator-theoretic fixed-point viewpoint suggested in the paper's future-work section could make the memristive model scalable to large networks, but that scalability remains to be demonstrated.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look. The paper's core idea is that the Hodgkin-Huxley fast-slow mixed feedback and Amari's short-range excitation/long-range inhibition are both instances of one memristive mixed-feedback mechanism, and it builds a single CNN-like memductance model that can do both. That is a real contribution: Eq. 11 is new, and the temporal model in Eq. 6 is simple and clean. The slow-fast derivation in Section IV-A, giving explicit excitability conditions (Eqs. 7-8), is the strongest part; it actually tells you why the model is excitable and it is robust in the sense that only inequalities matter. The authors deserve credit for turning an analogy into a concrete model architecture rather than just a remark.

The soft spots are where the reader and stress-test point. The spatial branch is the load-bearing half of the unification claim, and it is the least supported. Eq. 10 replaces Amari's signed kernel with two nonnegative conductance kernels feeding separate E and I populations. That is not a minor change; it is a different dynamical system, with a quadratic nonlinearity through g(v)(v-E) and no sigmoid firing-rate nonlinearity. No equivalence proof, no reduction, and no quantitative comparison to Amari's model is given. One handpicked simulation (Figure 8) shows a bump, but that does not establish that the model reproduces Amari's pattern-formation regimes. I also could not find the full equations for v_I; Eq. 10 defines the synaptic memductances, but the dynamics of the I population are not written out explicitly, which makes the simulation underspecified. No code or parameter sensitivity analysis is included, so it is hard to check how fragile the spatial behavior is.

The circularity worry is real but not disqualifying. The model is deliberately constructed so that the temporal currents mimic Na/K and the spatial currents mimic Amari's E/I balance; showing excitability under handpicked parameters is partly by construction. But that does not invalidate the architecture. The claim is that the mechanism is general, and the temporal conditions make that plausible for the temporal side. For the spatial side, the claim needs much more work.

This paper is for readers in computational neuroscience and control who care about unified models of excitability across scales. It deserves a serious referee, but it should not be accepted as is. The authors need to either prove (or at least clearly state) conditions under which the two-population memristive model reproduces Amari dynamics, or add systematic numerical comparison—bump widths, thresholds, stability, parameter sweeps—plus the missing v_I equations and code. If the spatial branch is fixed, this could be a nice reference. As it stands, it is a promising but preliminary framework.

Referee Report

4 major / 5 minor

Summary. The paper proposes a memristive model of spatio-temporal excitability that aims to unify Hodgkin-Huxley temporal excitability and Amari spatial excitability under a single mixed-feedback architecture. The authors revisit the HH and Amari models, represent their feedback structure as the difference of two currents, and introduce a conductance-based model in which temporal memductances (Section IV-A) and spatial memductances (Section IV-B) are implemented by simple CNN-type operators. Temporal excitability is characterized by explicit inequalities in Eqs. (7)-(8), and simulations in Figures 6, 8, and 9 illustrate temporal, spatial, and combined spatio-temporal responses. The central claim is that excitability at both scales is the result of one mechanism: fast/short-range positive feedback paired with slow/long-range negative feedback.

Significance. If fully substantiated, the proposed framework would offer a cross-scale modeling principle: the same RC-plus-memductance circuit structure would describe single-neuron and population excitability, with only the spatio-temporal kernel of the memductance changing. This is an attractive and potentially useful conceptual contribution for neuromorphic modeling and for linking cellular and population dynamics. The paper has notable strengths: the block-diagram analogy between HH and Amari is clearly drawn; the temporal model yields explicit, simple excitability conditions in Eqs. (7)-(8); and the parameter tables make the simulations reproducible in principle. However, the spatial branch, which is load-bearing for the unification claim, is validated only by a single handpicked simulation (Figure 8) with no equivalence proof, no quantitative comparison to Amari dynamics, and no sensitivity or stability analysis. The paper is therefore best read as a promising conceptual proposal rather than a demonstrated unification.

major comments (4)

[Section IV-B, Eq. (10), Figure 8] The spatial branch of the model is not shown to reproduce Amari dynamics. Eq. (10) replaces the signed Mexican-hat kernel w(x)=e^{-|x|/sigma_E}-e^{-|x|/sigma_I} of Eq. (4) with two nonnegative memductance kernels acting on separate excitatory and inhibitory populations with voltages v_E and v_I. This is not a change of variables or a singular limit of Amari's equation: the convolution is applied to the filtered voltage v rather than to a saturating firing rate f(u), the nonlinearity in the current is quadratic in v through g(v)(v-E), and there is no sigmoid firing-rate function. The claim that this two-population conductance model preserves Amari's spatial excitability therefore requires either an equivalence proof or a systematic quantitative comparison (e.g., threshold amplitude, bump width, stability, and parameter sweeps against Eq. (4)). Figure 8 shows one simulation only, which is insufficient support for the spatial half of the unification.
[Section IV-B, Eq. (10), Figures 8-9] The spatial model is underspecified. Eq. (10) defines the synaptic memductances g_syn through temporal filtering, spatial convolution, and a ReLU threshold, but the paper does not write the full coupled equations for the two populations v_E and v_I. The text says each population is connected to itself and to the other, but the exact right-hand sides of partial differential equations for both populations, the boundary conditions, the spatial domain, and the numerical discretization are not given. Without these details, the simulations in Figures 8 and 9 cannot be reproduced or independently checked. The authors should provide the complete PDE system and the numerical implementation details.
[Section V, Eq. (11)] The unified spatio-temporal model is introduced by simply adding the temporal and spatial memductances in Eq. (11), but no analysis or systematic test shows that the combined system preserves both excitability types. In particular, it is not shown that a spatially superthreshold input produces the same localized bump in the presence of the temporal currents, nor that a temporally superthreshold input still produces an action-potential-like response in the presence of the spatial currents. Figure 9 is a single simulation and does not quantify threshold separation or the interaction between the two mechanisms. The authors should demonstrate both excitability types in the combined model, and ideally characterize how the temporal and spatial memductances interact.
[Section IV-A, Eqs. (7)-(8)] The excitability conditions in Eqs. (7)-(8) are derived under the approximations v_e,m approximately v and v_i,m approximately 0, which are plausible for fast and slow time constants but are not rigorously justified. The paper states that the mechanism is 'robust to uncertainty in the parameter values' immediately after Eq. (8), yet no sensitivity analysis, parameter-space exploration, or basin/threshold study is provided. Since these conditions are central to the temporal excitability claim, the approximations and the robustness statement should be made precise, or at least supported by numerical experiments over the parameter ranges in Table III.

minor comments (5)

[Section IV-B, text before Eq. (9)] Typo: 'obain' should be 'obtain'.
[Table III caption] Typo: 'Whever applicable' should be 'Wherever applicable'.
[Section VI-B] Typo: 'significant' is misspelled as 'sigificant'.
[Tables III-IV and simulation section] The parameter tables give numerical values without units or a statement that the model is nondimensionalized; please clarify the units or state explicitly that all quantities are in arbitrary dimensionless units.
[Section VII] The simulation details state when steady-state values are read off (e.g., t=205 and t=305 for Figure 4), but the total simulation time and the spatial domain size are not given; please specify these to ensure reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the model's feedback conditions and simulations are derived from its own stated equations; the lack of a quantitative Amari-equivalence proof is an evidential weakness, not circularity.

full rationale

The paper's central derivation chain is self-contained. The temporal excitability conditions in Equations (7)-(8) are derived directly from the proposed memristive current-balance model, not from the Hodgkin-Huxley equations it is meant to unify. The spatial model in Equations (9)-(10) explicitly introduces its own nonnegative memductance kernel and two-population architecture; the resulting figures are qualitative demonstrations, not fitted predictions relabeled as confirmations. No parameter is fitted to Amari or Hodgkin-Huxley data and then reported as a predicted observable. The self-citations ([10], [11], [13], [14], [15]) appear only as motivation or as future-work references and are not load-bearing for the paper's main claim. The absence of a formal equivalence proof or systematic comparison between Equation (10) and Amari's Equation (4) is a substantive validation gap, but it does not make the derivation circular: the conclusion that the proposed model exhibits spatio-temporal excitability is read off from its own simulated dynamics, not imported from the cited models or from the target conclusion itself.

Assumptions & free parameters 19 free parameters · 6 assumptions · 0 invented entities

The model introduces no new physical entities. It rests on 19 hand-chosen parameters and several domain or ad hoc assumptions about what constitutes excitability and how to convert Amari's signed kernel into memristive form. The central claim is therefore a designed construction rather than a parameter-free derivation.

free parameters (19)

leak conductance g_l = 0.1
Sets passive membrane time constant and shifts the excitability thresholds in Eqs 7-8; chosen by hand in Tables III and IV.
fast excitatory memductance time constant tau_e,m = 0.1
Ensures fast positive feedback in the temporal excitability model, Table III.
max excitatory memductance g_e_bar = 1
Sets the strength of the positive feedback current, Table III.
excitatory reversal potential E_e = 10
Makes the excitatory current inward near rest, satisfying the positive feedback condition Eq 7, Table III.
excitatory memductance threshold vth_e,m = 1
Controls the activation threshold of the excitatory current, Table III.
slow inhibitory memductance time constant tau_i,m = 10
Ensures slower negative feedback than the excitatory current, Table III.
max inhibitory memductance g_i_bar = 10
Sets the strength of the negative feedback current, chosen larger than g_e_bar to satisfy Eq 8, Table III.
inhibitory reversal potential E_i = -10
Makes the inhibitory current outward, providing negative feedback, Table III.
inhibitory memductance threshold vth_i,m = 1
Controls the activation threshold of the inhibitory current, Table III.
excitatory spatial scale sigma_E = 0.5
Sets the short range of excitatory spatial interactions, Table IV.
excitatory synaptic time constant tau_E_syn = 0.1
Makes synaptic excitation faster than inhibition, Table IV.
excitatory synaptic threshold vth_E = 2
Sets the threshold for the excitatory synaptic memductance, Table IV.
max excitatory synaptic memductance g_E_syn_bar = 10
Sets the strength of the excitatory synaptic interaction, Table IV.
excitatory synaptic reversal potential E_E_syn = 10
Makes the excitatory synaptic current inward, Table IV.
inhibitory spatial scale sigma_I = 5
Sets the long range of inhibitory spatial interactions, chosen larger than sigma_E, Table IV.
inhibitory synaptic time constant tau_I_syn = 1
Makes synaptic inhibition slower than excitation, Table IV.
inhibitory synaptic threshold vth_I = 2
Sets the threshold for the inhibitory synaptic memductance, Table IV.
max inhibitory synaptic memductance g_I_syn_bar = 3
Sets the strength of the inhibitory synaptic interaction, Table IV.
inhibitory synaptic reversal potential E_I_syn = -10
Makes the inhibitory synaptic current outward, Table IV.

assumptions (6)

domain assumption The mixed-feedback motif of fast positive feedback plus slower negative feedback fully captures Hodgkin-Huxley temporal excitability.
Invoked in Section II and used to justify replacing four HH conductances with one excitatory and one inhibitory memductance in Equation 6.
domain assumption Amari's spatial excitability is fully captured by short-range excitation and long-range inhibition.
Invoked in Section III and used to design the spatial memductance kernels in Equation 10.
domain assumption A neuron or neural population can be represented as an RC circuit with parallel Ohmic current sources whose conductances have memory.
Core modeling principle retained from Hodgkin-Huxley and stated in Sections I and IV.
ad hoc to paper The memristive spatial kernel must be nonnegative, so excitation and inhibition must be modeled by separate populations.
Introduced in Section IV-B to convert Amari's signed kernel into Ohmic memductances; no derivation is given that the two-population model replicates Amari's dynamics.
ad hoc to paper A simple CNN operator with one temporal or spatial convolution per current is sufficient to capture excitability.
Stated in Section IV and demonstrated only for the specific parameter sets in Tables III and IV.
ad hoc to paper Temporal and spatial memductances can be combined additively in Equation 11 without losing either excitability type.
Section V simply adds the temporal and spatial currents; only one combined simulation is shown.

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Cite this review

Pith. "Pith review of A memristive model of spatio-temporal excitability." pith.science (2026). https://pith.science/paper/DO7OZZNG

@misc{pith2026250522269,
  author       = {Pith},
  title        = {Pith review of: A memristive model of spatio-temporal excitability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DO7OZZNG}},
  note         = {Machine review of arXiv:2505.22269}
}

read the original abstract

This paper introduces a model of excitability that unifies the mechanism of an important neuronal property both in time and in space. As a starting point, we revisit both a key model of temporal excitability, proposed by Hodgkin and Huxley, and a key model of spatial excitability, proposed by Amari. We then propose a novel model that captures the temporal and spatial properties of both models. Our aim is to regard neuronal excitability as a property across scales, and to explore the benefits of modeling excitability with one and the same mechanism, whether at the cellular or the population level.

Figures

Figures reproduced from arXiv: 2505.22269 by the authors.

**Figure 2.** Block diagram of the mixed temporal monotone structure of the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Sketch of w(x) as used in Amari’s model. of activity is mediated through a zoo of neurons, which can be broadly characterized into two classes: excitatory (E) and inhibitory (I). Excitatory neurons promote further neuronal activity, whilst inhibitory neurons usually diminish it. Excitation and inhibition are spatially structured, with E and I operating in a balanced regime. This can lead to interesting spatial patte… view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: Block diagram of the mixed spatial monotone structure of Amari’s [PITH_FULL_IMAGE:figures/full_fig_p003_5.png]

**Figure 6.** Figure 6: Temporal excitability of the memristive neuron model. A small [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 7.** Figure 7: Relationship between vE , vI and ge, gi to model spatial excitability. B. A model of spatial excitability Amari’s model exhibits the same mixed-feedback structure as the Hodgkin-Huxley model. However it is not memristive, that is, the feedback currents are not Ohmic. T…

**Figure 8.** Figure 8: Spatial excitability in the CNN model. Left: the subthreshold [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: Spatio-temporal excitability in the canonical model. The response [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

The Cognitive Correlates of Human Brain Oscilla- tions,

M. J. Kahana, “The Cognitive Correlates of Human Brain Oscilla- tions,”Journal of Neuroscience, vol. 26, pp. 1669–1672, Feb. 2006

work page 2006

[2] [2]

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[3] [3]

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J. Keener and J. Sneyd, “Mathematical physiology 1: Cellular physi- ology,” 2009

work page 2009

[4] [4]

E. M. Izhikevich,Dynamical systems in neuroscience. MIT press, 2007

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B. Ermentrout and D. H. Terman,F oundations of mathematical neuroscience. Citeseer, 2010

work page 2010

[6] [6]

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F. Buchholtz, J. Golowasch, I. R. Epstein, and E. Marder, “Mathemat- ical model of an identified stomatogastric ganglion neuron,”Journal of neurophysiology, vol. 67, no. 2, pp. 332–340, 1992

work page 1992

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H. R. Wilson and J. D. Cowan, “Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons,”Biophysical Journal, vol. 12, pp. 1–24, Jan. 1972

work page 1972

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S.-i. Amari, “Dynamics of pattern formation in lateral-inhibition type neural fields,”Biological Cybernetics, vol. 27, pp. 77–87, June 1977

work page 1977