Do all five models receive the same current?

They receive the exact same normalized command samples at the same simulation times. Because their native input units differ, the command passes through a fixed, visible gain for each model; those gains are never hidden auto-calibration.

Which neuron model is the most realistic?

Hodgkin–Huxley contains the most membrane biophysics in this comparison, but more detail is not automatically better for every question. A reduced model can be more useful when timing, logic, phase geometry, or computational scale is the object of study.

Why is biological energy unavailable for four models?

McCulloch–Pitts, LIF, Izhikevich, and FitzHugh–Nagumo do not resolve the ionic currents needed to infer an ion-restoration burden. Their answer is not zero; it is not identifiable from their state equations.

Does the Hodgkin–Huxley energy number equal a neuron's total energy use?

No. It is a baseline-corrected ATP-equivalent estimate for restoring inward sodium charge in canonical squid giant-axon membrane. It omits geometry, synapses, calcium, glia, maintenance, transmitter cycling, and most cellular metabolism.

The Neuron Zoo | Suvro Ghosh

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Five aligned neuron-model traces progressing from a binary threshold signal to a Hodgkin–Huxley action potential — One command enters five mathematical animals. Their disagreement is the lesson: each model preserves a different part of neuronal behaviour.

A “neuron” can be a yes-or-no gate, a leaking bucket with a bell, a two-variable conjurer, an excursion through a phase plane, or a membrane whose sodium and potassium gates open in time. Those descriptions are not interchangeable. Each is a decision about what one is willing not to know.

The laboratory below turns that decision into a controlled experiment. It constructs one normalized command waveform, samples it on one fixed clock, and gives the identical sample sequence to all five models. Their native input gains remain visible because a unitless logical drive, picoamperes, dimensionless forcing, and current density are not secretly the same physical quantity.

Change a preset, draw a command, shorten the paired-pulse interval, or move the synchronized data cursor. The plots retain their honest native axes: a binary output stays binary, a FitzHugh–Nagumo variable does not acquire fictional millivolts, and a leaky integrate-and-fire event does not grow a decorative action-potential waveform.

Interactive computational neuroscience

The Neuron Zoo

From Leaky Buckets to Hodgkin–Huxley

Send exactly the same normalized command into five neuron models and watch what each simplification keeps, invents, approximates, or throws overboard.

One current command. Five levels of biological detail.

01 MP 1943

02 LIF 1907 lineage

03 IZH 2003

04 FHN 1961–1962

05 HH 1952

Preparing the deterministic laboratory…

same s[k] one clock native axes module Worker

Preset Presentation speed

Pointer mode

Shared clock 0.000 ms / 1,000 ms fixed Δt 0.025 ms

Canonical command

Shared normalized stimulus, s(t)

+1 −1 · 1,000 ms

Drag to paint. Missed samples are filled by deterministic linear interpolation. Keyboard: arrows edit the selected sample; Shift makes a larger move.

One source of truth 40,000 Float64 command samples

Exact fixed clock Δt 0.025 ms · display 0.1 ms

Seed 1,337

No auto-calibration five visible native gains

Guided discoveries

Try this

Advanced settings and reproducibility

Changing the time step, duration, seed, phenotype, or input gain resets and recomputes the experiment. Presentation speed never changes the scientific time step.

Fixed integration step Experiment duration Noise seed Display sample interval Izhikevich phenotype FHN time scale, ms Assumed ΔG ATP, kJ/mol

Visible native-input gains

One command shape, translated into five incompatible native input spaces.

McCulloch–Pitts

unitless × s(t)

LIF

pA × s(t)

Izhikevich

model units × s(t)

FitzHugh–Nagumo

dimensionless × s(t)

Hodgkin–Huxley

µA/cm² × s(t)

Import experiment JSON

0.1 ms warning: this coarser step is offered for inspection, not as the validated default. Fixed-step approximation error can shift spike timing and ionic-charge estimates.

Five mathematical animals

The comparison wall

Primary Compare with

1943 · Binary threshold logic

McCulloch–Pitts

State: y
Native drive: 1 unitless threshold drive × s(t)

binary output y 0 ms

Keepsthreshold logic and all-or-none output

Throws awaymembrane, time-continuous recovery, channels, energetics

threshold logic: explicit
membrane voltage: unavailable
biological energy: unavailable

Equation and exact defaults

y[k+1] = H(u[k] + b - theta) | H(x) = 1 if x >= 0; otherwise 0

gain 1
bias 0
threshold 0.5

1907 lineage · Threshold-and-reset membrane model

Leaky integrate-and-fire

State: V, refractory time remaining
Native drive: 500 pA × s(t)

subthreshold membrane voltage V 0 ms

Keepspassive membrane, leak, threshold timing

Throws awaygenerated spike waveform, channels, physical ion energetics

passive leak: explicit
reset + timer: explicit
spike waveform: unavailable

Equation and exact defaults

C_m dV/dt = -g_L (V - E_L) + I(t) | V[k+1] = E_L + (V[k]-E_L) exp(-dt/tau_m) + I[k]/g_L (1-exp(-dt/tau_m))

Cₘ 200 pF
gL 10 nS
EL -65 mV
Vth -50 mV
Vreset -65 mV
tref 2 ms
input gain 500 pA

2003 · Two-variable hybrid spiking model

Izhikevich

State: v, u
Native drive: 15 model input units × s(t)

phenomenological voltage-like variable v 0 ms

Keepsspike timing, adaptation, bursting repertoire

Throws awayexplicit ion channels and physical current accounting

adaptation: phenomenological
hybrid reset: explicit
ion gates: unavailable

Equation and exact defaults

dv/dt = 0.04v^2 + 5v + 140 - u + I(t) | du/dt = a(bv-u) | if v >= 30 mV: v = c; u = u + d

phenotype regular spiking
a 0.02
b 0.2
c -65
d 8
input gain 15 model units

1961–1962 · Two-variable excitable phase-plane model

FitzHugh–Nagumo

State: v, w
Native drive: 0.8 dimensionless drive × s(t)

fast activation-like variable v 0 ms

Keepsexcitability, recovery, oscillation, phase geometry

Throws awayphysical channels, conductances, and ion accounting

recovery variable: explicit
refractory recovery: emergent
physical voltage: unavailable

Equation and exact defaults

dv/dt = (v - v^3/3 - w + I(t)) / T_FHN | dw/dt = epsilon (v + a - bw) / T_FHN

a 0.7
b 0.8
ε 0.08
TFHN 12.5 ms
input gain 0.8

1952 · Conductance-based membrane model

Hodgkin–Huxley

State: V, m, h, n
Native drive: 20 µA/cm² × s(t)

membrane voltage V 0 ms

Keepsvoltage-dependent channels, generated spike waveform, ion-current bookkeeping

Throws awaymorphology, spatial propagation, channel noise, synapses, and most cellular metabolism

ion gates: explicit
refractory recovery: emergent
sodium burden: defined

Equation and exact defaults

C_m dV/dt = I_ext - I_Na - I_K - I_L  |  I_Na = g_Na m^3 h (V-E_Na)  |  I_K = g_K n^4 (V-E_K)  |  I_L = g_L (V-E_L)  |  dx/dt = alpha_x(V)(1-x) - beta_x(V)x for x in {m,h,n}

Cₘ 1 µF/cm²
ḡNa 120 mS/cm²
ḡK 36 mS/cm²
gL 0.3 mS/cm²
input gain 20 µA/cm²

One horizontal time scale

Synchronized scope

The white cursor reports one simulation time across unlike, separately labelled axes.

Shared normalized command 0 ms · 0 unitless s(t)

McCulloch–Pitts 0 ms

Leaky integrate-and-fire 0 ms

Izhikevich 0 ms

FitzHugh–Nagumo 0 ms

Hodgkin–Huxley 0 ms

Spike and event raster 0 ms

Data view

Values at the synchronized time cursor

Time: 0 ms
Shared command: 0 unitless s(t)

Current state and key metrics for every neuron model at 0 milliseconds
Model	Current state	Key metrics
No model values are available at this cursor position.

Variables behind the primary trace

Detailed inspectors

Izhikevich recovery

Recovery variable u 0 ms

Hybrid reset

dv/dt = 0.04v² + 5v + 140 − u + I; du/dt = a(bv − u); at v ≥ 30, v ← c and u ← u + d.

FitzHugh–Nagumo phase plane

FitzHugh–Nagumo phase plane v 0 · w 0

The cubic nullcline uses the current command sample; v and w are dimensionless.

Hodgkin–Huxley gates

Hodgkin–Huxley gates 0 ms · m unavailable · h unavailable · n unavailable

Sodium current INa 0 ms

Potassium current IK 0 ms

Paired-pulse experiment

Refractory is not one universal stopwatch

Inter-pulse interval 12.0 ms

McCulloch–Pitts

absent

No internal refractory state in this single logical unit.

Recovery state: binary state OFF
Second pulse: 0 new rising edge(s)

LIF

explicit

A fixed timer holds voltage at reset after an event.

Recovery state: unavailable ms remaining
Second pulse: no event in 30 ms

Izhikevich

phenomenological

Reset and the evolving recovery variable u shape readiness.

Recovery state: u = unavailable
Second pulse: no event in 30 ms

FitzHugh–Nagumo

emergent

The slow variable w carries the trajectory back through phase space.

Recovery state: w = unavailable
Second pulse: no event in 30 ms

Hodgkin–Huxley

emergent

Sodium availability h and potassium activation n recover continuously.

Recovery state: h unavailable, n unavailable
Second pulse: no event in 30 ms

The response to the second pulse is the comparison. The laboratory does not manufacture one cross-model “refractory milliseconds” score.

What can be counted honestly?

Ion restoration is not computer cost

Hodgkin–Huxley only

Excess inward Na⁺ charge unavailable nC/cm² stimulated run minus equal-duration quiet baseline

ATP-equivalent burden unavailable molecules/cm² three sodium ions restored per ATP hydrolysed

ATP-equivalent amount unavailable mol/cm² using F = 96485.33212 C/mol

Raw charge and optional chemical-work assumption

Raw stimulated inward sodium: unavailable nC/cm²
Quiet shadow baseline: unavailable nC/cm²
Assumed ΔG ATP: 50 kJ/mol
Secondary chemical-work estimate: unavailable J/cm²

Reduced models

Biological energy: not identifiable from the McCulloch–Pitts, LIF, Izhikevich, or FitzHugh–Nagumo state equations. The answer is not zero.

Interpretation boundary

This is an ATP-equivalent sodium-restoration estimate, not total neuronal metabolism. It omits membrane area and geometry, synapses, calcium, transmitter cycling, glia, protein turnover, and maintenance beyond the chosen baseline. The membrane is canonical squid giant axon, not a complete human neuron.

Separate measurement

Computer execution cost

Model	States	Derivative evaluations/step	Median run, ms	Relative to LIF
McCulloch–Pitts	1	0	unavailable	unavailable
Leaky integrate-and-fire	2	0	unavailable	unavailable
Izhikevich	2	4	unavailable	unavailable
FitzHugh–Nagumo	2	4	unavailable	unavailable
Hodgkin–Huxley	4	4	unavailable	unavailable

What was thrown overboard?

Five different bargains

These phrases describe what each equation actually represents. “Not identifiable” means the requested quantity cannot be recovered from that model—not that its physical value is zero.

Scientific capability comparison for the five neuron models
Representation	1 McCulloch–Pitts	2 Leaky integrate-and-fire	3 Izhikevich	4 FitzHugh–Nagumo	5 Hodgkin–Huxley
Binary output	explicit 0/1 state	not represented	not represented	not represented	not represented
Continuous membrane-like variable	not represented	explicit V between resets	phenomenological v	dimensionless fast v	physical membrane V
Physical membrane voltage	not represented	explicit subthreshold mV	phenomenological voltage-like v	not represented	explicit mV
Passive leak	not represented	explicit gL	not separately represented	not separately represented	explicit ionic leak IL
Threshold event	explicit Heaviside threshold	explicit Vth crossing	explicit 30 mV cutoff	analysis convention: upward v = 1	analysis convention: upward V = 0 mV
Artificial reset	not represented	explicit Vreset	explicit v ← c; u ← u + d	none; trajectory continues	none; waveform continues
Explicit refractory timer	not represented	explicit tref	not represented	not represented	not represented
Recovery variable	not represented	timer only; no dynamic recovery variable	explicit phenomenological u	explicit slow w	emergent through h and n gates
Generated action-potential waveform	not represented	event only; no waveform	phenomenological hybrid waveform and reset	emergent dimensionless excursion	generated conductance-based voltage waveform
Ion-channel gates	not represented	not represented	not represented	not represented	explicit m, h, and n
Ionic currents	not identifiable	not identifiable	not identifiable	not identifiable	explicit INa, IK, and IL
Conductance parameters	not represented	leak conductance gL only	not represented	not represented	Na⁺, K⁺, and leak conductances
Phase-plane interpretation	not represented	one-dimensional subthreshold flow	phenomenological v–u geometry	explicit v–w nullcline geometry	four-dimensional; projections only
Physical biological-energy estimate	not identifiable from this model	not identifiable from this model	not identifiable from this model	not identifiable from this model	caveated sodium-restoration estimate
Native units	unitless command and binary state	ms, mV, pA, pF, nS	ms, mV-like v, model input units	dimensionless v, w, and model time; mapped to ms	ms, mV, µA/cm², mS/cm²
Typical computational cost	very low: threshold update	low: one state plus event logic	low: two states plus reset	moderate: two-state fixed-step RK4	highest here: four-state RK4 and rate functions

How to read the experiment

The single source of truth is a deterministic sample array

-1\le s(t)\le 1.

In plain text: the shared command, s of t, is bounded between minus one and plus one. Every model reads the same $s_k$ at the same time $t_k$ . The default experiment lasts

T=1000\ \text{ms}

and advances with the fixed internal step

\Delta t=0.025\ \text{ms}.

The advanced choices are 0.01, 0.025, 0.05, and 0.1 ms; the last carries an accuracy warning. Changing the time step resets and deterministically recomputes the run. Presentation speed changes how many fixed steps are calculated per second of wall time, never the scientific $\Delta t$ .

The command becomes a native drive through these visible defaults:

Model	Native input	Default mapping
McCulloch–Pitts	Unitless threshold drive	$u_{MP}=1.0\,s(t)$
Leaky integrate-and-fire	pA	$I_{LIF}=500\,s(t)\ \text{pA}$
Izhikevich	Model input units	$I_{IZH}=15\,s(t)$
FitzHugh–Nagumo	Dimensionless drive	$I_{FHN}=0.8\,s(t)$
Hodgkin–Huxley	$\mu\text{A}/\text{cm}^2$	$I_{HH}=20\,s(t)\ \mu\text{A}/\text{cm}^2$

These gains put one command shape into five useful operating ranges. They are defaults, not universal biological constants, and the laboratory does not tune them behind the scenes to force equal spike counts.

The deterministic stimulus shelf

Every named preset is generated from duration, time step, parameters, and—where noise is used—a seed:

Preset	Exact command
Quiet	$s(t)=0$
Single pulse	0 before 100 ms, +0.80 from 100 through 110 ms, then 0
Sustained step	0 before 100 ms, +0.55 from 100 through 700 ms, then 0
Pulse train	+0.75 for 10 ms at 100, 180, 260, 340, 420, and 500 ms
Ramp	0 before 100 ms, linear 0 to +0.90 from 100 to 600 ms, then 0
Paired pulse	Two +0.85 pulses, 8 ms wide; first at 100 ms, second after an adjustable 1–100 ms interval, default 12 ms
Hyperpolarize and release	-0.50 from 100 to 300 ms, +0.60 from 300 to 320 ms, then 0
Seeded noisy step	From 100 to 700 ms, base +0.42 plus filtered noise of scale 0.12; default seed 1337

For the noisy step, bounded uniform samples $r_k$ pass through a one-pole low-pass filter with a 2 ms time constant:

x_k=\alpha x_{k-1}+(1-\alpha)(2r_k-1), \qquad \alpha=e^{-\Delta t/(2\ \text{ms})},

s_k=\operatorname{clip}(0.42+0.12x_k,-1,1).

In plain text: each noise state retains the fraction alpha of its previous value, adds the remaining fraction of a new seeded sample mapped to minus one through plus one, then the final command is clipped to the shared range. The same seed, duration, and time step produce the same waveform and hash.

Five specimens, five kinds of answer

McCulloch–Pitts — 1943

A neuron as logic: add, compare, answer yes or no.

The scalar comparison used here is a modern single-input form of the McCulloch–Pitts threshold idea:

y_{k+1}=H(g_{MP}s_k+b-\theta),

H(x)= \begin{cases} 1, & x\ge 0,\\ 0, & x<0. \end{cases}

In plain text: the next output is one when gain times command plus bias reaches the threshold, and zero otherwise. Defaults are $g_{MP}=1.0$ , $b=0$ , $\theta=0.5$ , and $y_0=0$ .

This is not a membrane model. It has no voltage, leak, reset voltage, ion gates, physical refractory period, ionic current, or biological-energy quantity. A sustained suprathreshold command therefore holds the output at 1; turning that plateau into a fabricated train of biological spikes would change the model.

Keeps: threshold logic and all-or-none output.
Throws away: membrane, time-continuous recovery, channels, and energetics.

Leaky integrate-and-fire — 1907 lineage

A charging membrane with a leak, a threshold, and a trapdoor reset.

The lumped-current equation is

C_m\frac{dV}{dt}=-g_L(V-E_L)+I(t).

Equivalently,

\tau_m\frac{dV}{dt}=-(V-E_L)+R_mI(t), \qquad \tau_m=\frac{C_m}{g_L}.

In plain text: capacitance times voltage change equals applied current minus leak current. The defaults are $C_m=200$ pF, $g_L=10$ nS, $E_L=-65$ mV, $V_{th}=-50$ mV, $V_{reset}=-65$ mV, and $t_{ref}=2$ ms. The initial voltage is $E_L$ , and $I(t)=500s(t)$ pA. The units remain consistent because $1\ \text{nS}\cdot1\ \text{mV}=1\ \text{pA}$ .

Because the input is held constant for one command sample, the implementation can advance the linear subthreshold membrane exactly:

V_{k+1}=E_L+(V_k-E_L)e^{-\Delta t/\tau_m} +\frac{I_k}{g_L}\left(1-e^{-\Delta t/\tau_m}\right).

At threshold, the model records an event, sets $V$ to $V_{reset}$ , and holds it there for the explicit refractory interval. The event is real within this model; a drawn excursion to +30 mV would not be.

Keeps: passive membrane, leak, and threshold timing.
Throws away: a generated spike waveform, channels, and physical ion energetics.

Izhikevich — 2003

Two variables and a reset reproduce a small zoo of firing patterns.

The continuous part is

\frac{dv}{dt}=0.04v^2+5v+140-u+I(t),

\frac{du}{dt}=a(bv-u).

When $v\ge 30$ mV, the hybrid event rule is

v\leftarrow c,\qquad u\leftarrow u+d.

In plain text: the fast variable $v$ is driven by a quadratic expression, the slower $u$ follows and opposes it, and reaching the cutoff triggers a reset of both variables. The regular-spiking default is $a=0.02$ , $b=0.20$ , $c=-65$ , $d=8$ , initial $v=-65$ , initial $u=bv=-13$ , and $I(t)=15s(t)$ in model input units.

Phenotype	$a$	$b$	$c$	$d$
Regular spiking	0.02	0.20	-65	8
Intrinsically bursting	0.02	0.20	-55	4
Chattering	0.02	0.20	-50	2
Fast spiking	0.10	0.20	-65	2
Low-threshold spiking	0.02	0.25	-65	2

The 30 mV cutoff belongs to this phenomenological event/reset convention; it is not an ion-channel-resolved summit. The variable $v$ is conventionally interpreted in mV under this parameterization, but the model is not a conductance-based membrane.

Keeps: spike timing, adaptation, and a bursting repertoire.
Throws away: explicit ion channels and physical current accounting.

FitzHugh–Nagumo — 1961–1962

Excitability becomes a map with a fast road and a slow road.

The laboratory uses this explicit dimensionless variant in model time $\tau$ :

\frac{dv}{d\tau}=v-\frac{v^3}{3}-w+I(t),

\frac{dw}{d\tau}=\epsilon(v+a-bw).

Defaults are $a=0.7$ , $b=0.8$ , $\epsilon=0.08$ , and $I(t)=0.8s(t)$ . Model time is tied visibly to the shared millisecond clock:

\tau=\frac{t}{T_{FHN}}, \qquad T_{FHN}=12.5\ \text{ms}.

The millisecond solver therefore integrates

\frac{dv}{dt}=\frac{1}{T_{FHN}}\left(v-\frac{v^3}{3}-w+I(t)\right),

\frac{dw}{dt}=\frac{\epsilon}{T_{FHN}}(v+a-bw).

In plain text: $v$ responds quickly, $w$ recovers slowly, and the visible 12.5 ms scale converts dimensionless model time to the common clock. At zero input the initial stable equilibrium satisfies

w=\frac{v+a}{b}, \qquad v-\frac{v^3}{3}-w=0,

giving $v_0\approx-1.199408035244035$ and $w_0\approx-0.624260044055044$ for the defaults.

There is no artificial reset. For event counting only, the laboratory records an upward crossing of $v_{event}=1.0$ . That is an analysis convention, not a hard threshold in the equations. Both $v$ and $w$ are dimensionless; neither is literal membrane voltage.

Keeps: excitability, recovery, oscillation, and phase geometry.
Throws away: physical channels, conductances, and ion accounting.

Hodgkin–Huxley — 1952

A spike assembled from voltage-dependent sodium and potassium conductances.

This implementation uses the modern absolute-voltage convention, an algebraically shifted form of the canonical squid giant-axon equations. All reversal potentials and rate functions use that convention consistently. With outward-positive ionic current,

C_m\frac{dV}{dt}=I_{ext}-I_{Na}-I_K-I_L,

I_{Na}=\bar g_{Na}m^3h(V-E_{Na}), \qquad I_K=\bar g_Kn^4(V-E_K), \qquad I_L=g_L(V-E_L).

In plain text: applied current charges the membrane after outward sodium, potassium, and leak currents are subtracted. The defaults are:

Quantity	Default
$C_m$	$1\ \mu\text{F}/\text{cm}^2$
$\bar g_{Na}$	$120\ \text{mS}/\text{cm}^2$
$\bar g_K$	$36\ \text{mS}/\text{cm}^2$
$g_L$	$0.3\ \text{mS}/\text{cm}^2$
$E_{Na}$	$50$ mV
$E_K$	$-77$ mV
$E_L$	$-54.387$ mV
Initial $V$	$-65$ mV
$I_{ext}(t)$	$20s(t)\ \mu\text{A}/\text{cm}^2$

For each gate $x\in\{m,h,n\}$ ,

\frac{dx}{dt}=\alpha_x(V)(1-x)-\beta_x(V)x.

The absolute-voltage rate functions are

\alpha_m(V)=\frac{0.1(V+40)}{1-e^{-(V+40)/10}}, \qquad \beta_m(V)=4e^{-(V+65)/18},

\alpha_h(V)=0.07e^{-(V+65)/20}, \qquad \beta_h(V)=\frac{1}{1+e^{-(V+35)/10}},

\alpha_n(V)=\frac{0.01(V+55)}{1-e^{-(V+55)/10}}, \qquad \beta_n(V)=0.125e^{-(V+65)/80}.

In plain text: every gate opens at rate alpha times its unavailable fraction and closes at rate beta times its current fraction. At $V=-65$ mV, each starts at

x_\infty(V)=\frac{\alpha_x(V)}{\alpha_x(V)+\beta_x(V)},

with $m_0\approx0.0529324852572496$ , $h_0\approx0.596120753508460$ , and $n_0\approx0.317676914060697$ . The apparent zero-over-zero points in $\alpha_m$ near -40 mV and $\alpha_n$ near -55 mV are removable singularities; the numerical implementation uses their analytic limits rather than raw division.

The model generates its own voltage waveform. A spike event is recorded when $V$ crosses 0 mV upward, but voltage is never reset. Recovery emerges from sodium inactivation and potassium activation.

Keeps: voltage-dependent channels, a generated spike waveform, and ion-current bookkeeping.
Throws away: morphology, spatial propagation, channel noise, synapses, and most cellular metabolism.

Refractory is not one universal stopwatch

The paired-pulse experiment reveals five different meanings hiding behind one familiar word:

Model	What recovery means here
McCulloch–Pitts	No internal refractory state; both suprathreshold pulses are represented
LIF	An explicit fixed timer; a pulse inside the 2 ms interval cannot trigger an event
Izhikevich	Reset plus the evolving recovery variable $u$ , with no default fixed timer
FitzHugh–Nagumo	Emergent return through $(v,w)$ phase space, with no hard reset
Hodgkin–Huxley	Emergent sodium availability $h$ and potassium activation $n$ , with no hard reset

The laboratory does not compress those mechanisms into one bogus “refractory milliseconds” ranking. The honest comparison is whether—and how—the second pulse changes the trajectory.

Biological energy is not computer cost

Only Hodgkin–Huxley resolves an ionic current from which this laboratory can estimate a sodium-restoration burden. Under the outward-positive convention, inward sodium is the negative part of $I_{Na}$ :

Q_{Na,in}=\int \max(-I_{Na}(t),0)\,dt.

With current density in $\mu\text{A}/\text{cm}^2$ and time in ms, $1\ \mu\text{A}\cdot\text{ms}=1\ \text{nC}$ , so the integral yields nC/cm². A shadow Hodgkin–Huxley run starts from the identical initial state, lasts the identical time, and receives zero command. The headline value is

Q_{excess}=\max(Q_{stimulus}-Q_{baseline},0).

Converting charge $Q_C$ in coulombs per cm² to an ATP-equivalent burden uses three sodium ions transported per ATP hydrolysed by the sodium–potassium pump:

n_{ATP}=\frac{Q_C}{3F}, \qquad N_{ATP}=n_{ATP}N_A,

where $F=96485.33212$ C/mol and $N_A=6.02214076\times10^{23}$ mol⁻¹. The optional secondary chemical-work estimate is

E=n_{ATP}\Delta G_{ATP},

with an explicitly assumed default $\Delta G_{ATP}=50$ kJ/mol and an advanced range of 45–60 kJ/mol.

This is an ATP-equivalent sodium-restoration estimate, not total neuronal metabolism. It assumes inward sodium must ultimately be pumped out. It omits cell area and geometry, synapses, calcium, transmitter cycling, resting maintenance beyond the selected baseline, glial work, protein turnover, and many other processes. The canonical membrane is squid giant axon, not a complete human neuron.

For McCulloch–Pitts, LIF, Izhikevich, and FitzHugh–Nagumo the correct display is: Biological energy: not identifiable from this model’s state equations. Zero would be a made-up result.

Worker execution time is reported separately, after warm-up and repeated median measurements, as computer cost. A cheap equation is not a metabolically efficient cell, and a biologically detailed equation does not burn ATP inside the browser.

Numerical methods and reproducibility

All continuous models advance on the same fixed time grid using Float64 arithmetic in a framework-independent numerical core. FitzHugh–Nagumo and Hodgkin–Huxley use fixed-step fourth-order Runge–Kutta. Izhikevich uses a stable fixed-step continuous update followed by its explicit event/reset rule. LIF uses the stable exact scalar update above. McCulloch–Pitts evaluates deterministically on the shared grid.

The default display interval is 0.1 ms—every fourth internal step—while spike crossings, extrema, gate bounds, ionic charge, and validation metrics are calculated from the full 0.025 ms stream. When a trace contains more samples than horizontal pixels, deterministic min/max envelopes preserve narrow peaks instead of blindly dropping every nth value.

The five models run in one versioned module Web Worker. requestAnimationFrame paints results but never defines simulation time. Run identifiers prevent a late frame from an earlier reset replacing the current experiment; substantial typed-array payloads are transferred rather than copied. Pausing an offscreen or hidden page pauses presentation and does not create a wall-clock catch-up burst.

A replay is therefore a numerical claim: the same preset or recorded waveform, seed, duration, time step, phenotype, gains, and parameters must reproduce the same command hash, event times within stated tolerances, and trace hashes. If a parameter choice becomes unstable, the instrument pauses, preserves the last valid result, explains the problem, and offers the exact defaults. It does not massage a NaN into a pretty line.

What was thrown overboard?

Model	Primary output	Reset and recovery	Physical ion accounting
McCulloch–Pitts	Binary, unitless state	Not represented in this single logical unit	Not identifiable
LIF	Subthreshold voltage in mV plus event ticks	Explicit reset and refractory timer	Not identifiable
Izhikevich	Phenomenological $v$ plus recovery $u$	Hybrid reset; recovery through $u$	Not identifiable
FitzHugh–Nagumo	Dimensionless fast $v$ and slow $w$	Emergent phase-plane recovery	Not identifiable
Hodgkin–Huxley	Physical membrane voltage in mV	Emergent through $m$ , $h$ , and $n$ ; no reset	Explicit membrane current densities; partial sodium-restoration estimate

There is no winner hiding in the final column. McCulloch–Pitts is useful when logic is the question. LIF is useful when event timing and large networks matter. Izhikevich buys a broad firing repertoire with two variables. FitzHugh–Nagumo makes excitability and recovery geometrically legible. Hodgkin–Huxley pays more computation for conductances, gates, currents, and a generated spike.

Assumptions, accessibility, and limits

This is an educational single-compartment comparison. It omits dendrites, axonal propagation, synaptic networks, spatial cable effects, stochastic channel noise, changing ion concentrations, detailed pumps, morphology, and the metabolic life surrounding the membrane. The five models do not describe the same biological object at the same level.

The synchronized plots use separately labelled native axes; no essential distinction depends on colour. The data cursor works by keyboard as well as pointer, and the Data view exposes current time, command, native inputs, states, events, extrema, and paired-pulse response in a textual table. Important changes are announced sparingly. Reduced-motion mode begins paused, and the equations and prose remain useful without animation.

Long equations may scroll within their own panels on small screens. Each equation is paired with a plain-language or plain-text statement of its variables and units. The plotted sample count is not an accessible summary; the current values and scientifically relevant metrics are.

Primary papers and historical source

Warren S. McCulloch and Walter Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity”, The Bulletin of Mathematical Biophysics 5 (1943), 115–133.
Louis Lapicque, “Quantitative Investigations of Electrical Nerve Excitation Treated as Polarization”, 1907 paper in English translation; historical context in Nicolas Brunel and Mark C. W. van Rossum, “Lapicque’s 1907 Paper: From Frogs to Integrate-and-Fire”, Biological Cybernetics 97 (2007), 337–339.
Alan L. Hodgkin and Andrew F. Huxley, “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”, The Journal of Physiology 117 (1952), 500–544.
Richard FitzHugh, “Impulses and Physiological States in Theoretical Models of Nerve Membrane”, Biophysical Journal 1 (1961), 445–466.
Jinichi Nagumo, Suguru Arimoto, and Shuji Yoshizawa, “An Active Pulse Transmission Line Simulating Nerve Axon”, Proceedings of the IRE 50 (1962), 2061–2070.
Eugene M. Izhikevich, “Simple Model of Spiking Neurons”, IEEE Transactions on Neural Networks 14 (2003), 1569–1572.

The equations above state the exact variants implemented here. Historical papers often use different notation or voltage origins; especially for Hodgkin–Huxley, mixing the original resting-relative convention with modern absolute voltages would produce a different and incorrect model.

Quick Answer

Key Terms

Do all five models receive the same current?

Which neuron model is the most realistic?

Why is biological energy unavailable for four models?

Does the Hodgkin–Huxley energy number equal a neuron's total energy use?

Shared normalized stimulus, s(t)

Try this

Visible native-input gains

The comparison wall

Synchronized scope

Detailed inspectors

Izhikevich recovery

FitzHugh–Nagumo phase plane

Hodgkin–Huxley gates

Refractory is not one universal stopwatch

Ion restoration is not computer cost

Computer execution cost

Word Cloud

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