A membrane behaves like a leaky capacitor

Separate positive and negative charge across a thin insulating membrane and you obtain a capacitor. The voltage across it changes when net current arrives. Lipid is the insulator; ion channels are selective conductors; concentration gradients and electric fields jointly determine the potential towards which each ionic current pulls.

For a compact single compartment, the bookkeeping begins with:

The term g(E - V) is the key. Conductance g says how open the route is. The driving force E - V says how strongly ions move through it at the present voltage. Opening the same number of channels therefore need not deliver the same current twice. As voltage approaches the reversal potential, the current shrinks. Cross the reversal potential and the current changes direction.

Current injection and synaptic conductance ask different questions

An ideal current source supplies the requested current regardless of the membrane voltage. A synaptic conductance does not. Its current is large when voltage is far from the reversal potential and weak when voltage is near it. This distinction explains why two laboratory stimuli that produce the same initial deflection can diverge later.

Suppose a current pulse and an excitatory conductance pulse are tuned to produce the same 5 mV rise at rest. As the membrane depolarises, the current source keeps pushing with its prescribed amplitude. The conductance pulse loses driving force because the voltage is approaching its reversal potential. Add an inhibitory conductance and the two stimuli separate further: the external current is partly shunted, while the excitatory synaptic current is both shunted and reduced by its own changing driving force.

This is why a model fitted under current-clamp stimulation need not predict conductance-driven synaptic input. The input interface is part of the mechanism. Replacing conductance with injected current is an intervention, not a neutral change of notation. The replacement can be justified when voltage stays in a narrow range and the resulting current error is negligible for the decision. It fails when depolarisation, reversal, saturation or concurrent inhibition is precisely what the experiment probes.

Why shunting inhibition can be a silent veto

Suppose an inhibitory conductance reverses near the resting voltage. Opening it alone causes little current because E_I - V is small. Yet total conductance increases, so the effective membrane resistance falls. When excitation arrives, part of its current leaves through the newly open inhibitory route. The membrane charges less and decays faster.

This is more than a linguistic correction. Mitchell and Silver found that shunting inhibition in cerebellar granule cells could divisively alter the input-output relation under particular conductance and noise regimes.^[5] It does not follow that every inhibitory synapse performs pure division. Location, reversal potential, background conductance and stimulus statistics matter. The safer conclusion is narrower: inhibition can change gain through conductance even when its immediate voltage signature looks small.

flowchart TD
  A["The inhibitory input that<br/>hardly moved the voltage"] --> B["The voltage that is not a vote"]
  B --> C["A spike is a trajectory"]
  C --> D["The branch changes the computation"]
  D -. "re-test" .-> A
  classDef first fill:#0a1424,stroke:#B96D00,stroke-width:2px,color:#e6f3ff
  classDef step fill:#0a1424,stroke:#3f7da0,stroke-width:1px,color:#e6f3ff
  class A first
  class B,C,D step

The threshold moves because the state moves

In a threshold unit, activation is usually computed from the present inputs and fixed parameters. The output changes when the weighted sum crosses a boundary. In an excitable membrane, the apparent boundary is produced by interacting state variables. Sodium activation, sodium inactivation and potassium activation evolve at different rates. Their history determines whether a small depolarisation decays, triggers a spike or arrives during refractoriness.

Hodgkin and Huxley represented these processes with conductances controlled by gating variables. In compact form:

A small rise in voltage increases sodium activation. Inward sodium current then raises voltage further, creating regenerative positive feedback. Slower sodium inactivation and potassium activation reverse the movement. The spike is the resulting excursion through a state space. It is not a single comparison operation hidden inside a biological casing.^[1]

Threshold is a surface through hidden state

A voltage value such as minus 50 mV is often called “the threshold”. That shorthand is useful for spike detection, but it can mislead as a causal account. Picture two membranes at exactly minus 50 mV. In the first, many sodium channels are available and potassium activation is low. In the second, sodium channels are partly inactivated and potassium conductance remains elevated after a recent spike. The first state may accelerate into an action potential; the second may fall back.

In the full state space, voltage is one coordinate among several. The boundary separating returning trajectories from spiking trajectories is therefore a surface whose position changes with channel state, temperature, neuromodulation and recent input. A fixed voltage threshold projects that surface onto one axis. The projection can work well under a restricted regime while failing under adaptation or closely spaced stimulation.

This gives a clean role to simplified spiking models. An integrate-and-fire model keeps voltage-like state and an event boundary while replacing the spike waveform with a reset. Models such as Izhikevich's retain a small number of dynamical variables to reproduce multiple firing patterns at low computational cost.^[13] They are not halfway points on a universal realism scale. Each preserves a different set of trajectories. The right simplification is the one whose hidden state still distinguishes the histories your experiment will compare.

Real neurons distribute channel classes unevenly. In many cortical pyramidal neurons, the axon initial segment has a specialised channel arrangement that favours spike initiation. Kole and colleagues showed that high sodium-channel density there is required for action-potential generation in the neurons they studied.^[4] This matters because the neuron's decisive comparison is not necessarily made at the soma, and the path from dendrite to axon initial segment is itself stateful.

Move a synapse and you change its transfer function

Rall's cable-theoretic work made a basic fact mathematically tractable: voltage spreads through dendrites with attenuation and delay that depend on membrane and axial properties, branch diameters and path length.^[2] Two identical synapses on different branches therefore need not have identical influence at the soma or axon initial segment.

Passive attenuation is only the beginning. Dendrites can contain voltage-gated sodium, calcium, potassium and NMDA-receptor conductances. Local depolarisation can recruit these mechanisms, creating branch-specific amplification, plateaus or spikes. In rat pyramidal neurons, Branco and Häusser found systematic integration gradients: proximal inputs summed more nearly linearly and demanded tighter coincidence, while distal inputs showed stronger amplification over broader intervals in the preparation studied.^[7]

A branch can act like a conditional subunit

Poirazi, Brannon and Mel showed that a detailed model of a pyramidal neuron could be approximated, for the rate-based task they examined, by a two-layer network whose hidden units corresponded to dendritic subunits.^[8] This is a useful functional compression. It is not evidence that a dendrite literally contains an artificial hidden layer, nor that all neurons require the same abstraction.

Order can matter without changing the input set

Dendritic geometry also turns temporal order into a spatial variable. Branco, Clark and Häusser demonstrated sequence discrimination in cortical dendrites: activating the same set of locations in opposite orders could produce different somatic responses because propagation and local nonlinearities differed along the branch.^[6] A bag of active synapses cannot express this distinction. A model needs at least location, timing and the relevant propagation rule.

Human cortical neurons add further reasons for restraint. Studies of ex vivo human tissue have reported enhanced compartmentalisation and dendritic calcium action potentials with properties that differ from common rodent examples.^[10][9] These findings are important but tissue-, cell-type- and protocol-bounded. They do not licence a generic claim that “the human neuron” performs a single exotic operation.

Compartmentalisation can hide as well as amplify

Dendritic nonlinearity is often described as extra computing power, but isolation can also prevent a local event from dominating the soma. A branch may support a calcium or NMDA event that changes local integration or plasticity while producing only a modest somatic voltage. Other branch geometry can favour strong coupling. The same tree therefore contains several observation points with different answers to “did the neuron respond?”

This matters for measurement. A somatic electrode can miss a local dendritic event; a calcium signal can reveal local activation without proving that the axon changed its output; a spike count can preserve the final event while hiding which branch caused it. The candidate output must be named at the scale where the mechanism is claimed. A soma-to-spike model cannot be used to explain a branch-specific intervention merely because its final firing rate fits.

stateDiagram-v2
  [*] --> Claim
  state "A membrane behaves like a<br/>leaky capacitor" as Claim
  state "Current injection and synaptic<br/>conductance ask different<br/>questions" as Test
  state "Where will three conductances<br/>pull the voltage?" as Revision
  Claim --> Test: expose mechanism
  Test --> Decision: evidence survives
  Test --> Revision: rival survives
  Revision --> Claim: narrow or rebuild
  Decision --> [*]

The artificial neuron was designed to forget

McCulloch and Pitts deliberately abstracted nervous activity into threshold logic.^[3] Modern artificial units usually compute a weighted sum followed by a nonlinearity:

That forgetting is often the point. A network trained for image classification does not need to simulate sodium current to be useful. The danger begins when success on one projected relation is treated as evidence that the discarded mechanism has also been captured.

Beniaguev, Segev and London trained deep networks to approximate the input-output behaviour of a detailed cortical-neuron model.^[11] The result shows that a sufficiently expressive artificial network can emulate that configured model over sampled spatiotemporal inputs. It does not collapse the distinction between emulator and target. The mapping is evidence for functional approximation under a test distribution, not identity of substrate, state or counterfactual response.

Three meanings of “the same computation”

Claims of sameness often slide among three standards. Under decision equivalence, two systems return the same class, rate or action on a specified input set. Under trajectory equivalence, relevant intermediate states and timings also correspond. Under intervention equivalence, manipulating a named cause in either system changes the matched outcome in the corresponding way.

Decision equivalence is usually the cheapest to establish and often all an application needs. Trajectory equivalence is stronger but representation-dependent: two systems can reach the same decision through different paths. Intervention equivalence is the demanding standard for a mechanistic substitute because it asks the model to preserve what happens when the world, rather than only the observed input, is changed.

These standards should not be collapsed into a single similarity score. A model can have excellent decision equivalence and poor intervention equivalence, as Figure 10 will show. It can also preserve a useful intervention while using different microscopic variables, provided a tested mapping connects the manipulated cause and effect. “Same computation” becomes informative only after the equivalence standard and its test distribution are declared.

A five-level causal fidelity ladder

Instead of asking for a universally realistic neuron, match the model to the intervention burden. The following ladder is a practitioner synthesis developed here. Higher levels add obligations; they are not moral rankings, and a lower level may be the correct engineering choice.

Biological inspiration is not biological validation

Dendrite-inspired artificial architectures can be valuable even when they are not biophysical models. Recent work has reported accuracy, robustness or parameter-efficiency gains from structured dendritic components in particular artificial-network tasks.^[14] Such results support an architectural hypothesis: local subunits and sparse receptive fields can impose useful inductive bias. They do not show that the network has reproduced a cortical neuron's membrane mechanism.

The distinction protects both fields. Machine learning can borrow branch-like factorisation without pretending to simulate ion channels. Neuroscience can use artificial networks as approximators without treating trainability as a biological explanation. A responsible comparison names the preserved relation, such as local conjunction or sparse routing, then tests where the relation breaks.

How to read the lab

The current pulse preset adds a fixed external current. The excitatory conductance preset opens three decaying conductance pulses whose current falls as voltage approaches 0 mV. The silent veto adds inhibition with a reversal near rest before excitation. The late inhibition moves the same conductance after the first decisive excursion.

A positive result is not “the neuron is intelligent”. It is a visible dependency: changing conductance, reversal potential or onset changes the trajectory in a way a context-free input sum would not predict. A negative result means the chosen perturbation was too weak, mistimed or outside the model's sensitive regime. It does not show conductance is generally irrelevant.

The lab encodes a single isopotential compartment, classic squid-axon channel kinetics and synthetic synaptic pulses. It omits dendritic cable, stochastic release, temperature corrections, ion concentration dynamics, channel diversity and cell-type calibration. It is an intuition instrument, not a fitted model of a named cortical neuron.

from __future__ import annotations

from dataclasses import dataclass
from typing import NamedTuple

import matplotlib.pyplot as plt
import numpy as np


@dataclass(frozen=True)
class Config:
    duration_ms: float = 60.0
    dt_ms: float = 0.01
    current_uA_cm2: float = 0.0
    g_exc_mS_cm2: float = 0.06
    g_inh_mS_cm2: float = 0.20
    inhibition_onset_ms: float = 9.0
    e_exc_mV: float = 0.0
    e_inh_mV: float = -65.0


class Trace(NamedTuple):
    time_ms: np.ndarray
    voltage_mV: np.ndarray
    g_exc_mS_cm2: np.ndarray
    g_inh_mS_cm2: np.ndarray


PRESETS: dict[str, Config] = {
    "current": Config(current_uA_cm2=10.0, g_exc_mS_cm2=0.0, g_inh_mS_cm2=0.0),
    "excitation": Config(g_inh_mS_cm2=0.0),
    "silent_veto": Config(),
    "late_inhibition": Config(inhibition_onset_ms=18.0),
}


def _safe_ratio(x: float, scale: float) -> float:
    return scale if abs(x) < 1e-9 else x / (1.0 - np.exp(-x / scale))


def alpha_n(v: float) -> float:
    return 0.01 * _safe_ratio(v + 55.0, 10.0)


def beta_n(v: float) -> float:
    return 0.125 * np.exp(-(v + 65.0) / 80.0)


def alpha_m(v: float) -> float:
    return 0.1 * _safe_ratio(v + 40.0, 10.0)


def beta_m(v: float) -> float:
    return 4.0 * np.exp(-(v + 65.0) / 18.0)


def alpha_h(v: float) -> float:
    return 0.07 * np.exp(-(v + 65.0) / 20.0)


def beta_h(v: float) -> float:
    return 1.0 / (1.0 + np.exp(-(v + 35.0) / 10.0))


def exp_pulse(time: np.ndarray, onset: float, tau: float) -> np.ndarray:
    return np.where(time >= onset, np.exp(-(time - onset) / tau), 0.0)


def simulate(config: Config) -> Trace:
    """Run a deterministic, single-compartment Hodgkin-Huxley simulation."""
    time = np.arange(0.0, config.duration_ms + config.dt_ms, config.dt_ms)
    voltage = np.full(time.shape, -65.0)
    g_exc = config.g_exc_mS_cm2 * sum(
        exp_pulse(time, onset, 5.0) for onset in (10.0, 15.0, 20.0)
    )
    g_inh = config.g_inh_mS_cm2 * exp_pulse(
        time, config.inhibition_onset_ms, 8.0
    )

    m = alpha_m(-65.0) / (alpha_m(-65.0) + beta_m(-65.0))
    h = alpha_h(-65.0) / (alpha_h(-65.0) + beta_h(-65.0))
    n = alpha_n(-65.0) / (alpha_n(-65.0) + beta_n(-65.0))

    for i in range(1, time.size):
        v = voltage[i - 1]
        dt = config.dt_ms
        m += dt * (alpha_m(v) * (1.0 - m) - beta_m(v) * m)
        h += dt * (alpha_h(v) * (1.0 - h) - beta_h(v) * h)
        n += dt * (alpha_n(v) * (1.0 - n) - beta_n(v) * n)

        i_na = 120.0 * m**3 * h * (v - 50.0)
        i_k = 36.0 * n**4 * (v + 77.0)
        i_leak = 0.3 * (v + 54.387)
        i_syn = g_exc[i] * (config.e_exc_mV - v)
        i_syn += g_inh[i] * (config.e_inh_mV - v)
        i_external = (
            config.current_uA_cm2 if 10.0 <= time[i] < 40.0 else 0.0
        )

        voltage[i] = v + dt * (i_external + i_syn - i_na - i_k - i_leak)

    return Trace(time, voltage, g_exc, g_inh)


def spike_count(voltage: np.ndarray) -> int:
    return int(np.count_nonzero((voltage[:-1] < 0.0) & (voltage[1:] >= 0.0)))


def plot(trace: Trace, title: str) -> None:
    fig, axes = plt.subplots(2, 1, sharex=True, figsize=(9, 6))
    axes[0].plot(trace.time_ms, trace.voltage_mV)
    axes[0].set_ylabel("Voltage (mV)")
    axes[0].set_title(title)
    axes[1].plot(trace.time_ms, trace.g_exc_mS_cm2, label="gE")
    axes[1].plot(trace.time_ms, trace.g_inh_mS_cm2, label="gI")
    axes[1].set_xlabel("Time (ms)")
    axes[1].set_ylabel("Conductance (mS/cm²)")
    axes[1].legend()
    fig.tight_layout()
    plt.show()


if __name__ == "__main__":
    preset = "silent_veto"
    trace = simulate(PRESETS[preset])
    print(f"Preset: {preset}")
    print(f"Spikes: {spike_count(trace.voltage_mV)}")
    print(f"Peak voltage: {trace.voltage_mV.max():.2f} mV")
    plot(trace, "Conductance intervention")

sequenceDiagram
  participant Q as Question
  participant M as Mechanism
  participant E as Evidence
  participant D as Decision
  Q->>M: The inhibitory input that hardly moved the voltage
  M->>E: A membrane behaves like a leaky capacitor
  E-->>M: compare rival explanations
  E->>D: Escalate detail only when the abstraction breaks
  D-->>Q: revise scope or act

The picture of a neuron as a weighted sum followed by a threshold is neither false nor complete. It is a compressed answer to a particular question. It can preserve a useful decision boundary while omitting conductance, hidden channel state, spike trajectory, branch geometry and ionic context.

The practical mistake is to choose detail by resemblance. A model with biological labels may still miss the causal variable under study. A plain artificial unit may be exactly right for a stable classification task. The decisive test is whether the abstraction remains valid when the variable you intend to manipulate is changed.

For network optimisation, begin with the cheapest unit that achieves the operational objective. For spike timing, retain temporal state. For adaptation, channel perturbation or pharmacology, retain the relevant conductances and gates. For synapse relocation, local dendritic events or branch-specific plasticity, retain morphology. Escalate only when a lower model breaks the intervention contract.

This changes how biological and machine intelligence should be compared. Do not ask whether one artificial activation “is” one neuron. Ask which projection links them, what was deliberately discarded, and which counterfactual would expose that loss. That language makes strong analogies testable and modest abstractions defensible.

A neuron is not complicated because biology failed to discover a cleaner activation function. Its computation is the organised evolution of material state. An artificial activation becomes scientifically informative when we name the slice of that evolution it preserves and refuse to smuggle the rest back in.