single-entity-spiking-neuron-survey

name	single-entity-spiking-neuron-survey
description	Comprehensive survey of single-entity spiking neuron models covering mathematical formulations, biological plausibility, and computational trade-offs. Covers integrate-and-fire variants (LIF, EIF, Izhikevich, AdEx), Hodgkin-Huxley models, FitzHugh-Nagumo, Morris-Lecar, and discrete/continuous analogs for membrane potential dynamics. Activation: spiking neuron model, neuron model survey, LIF, EIF, Izhikevich, AdEx, Hodgkin-Huxley, FitzHugh-Nagumo, Morris-Lecar, membrane potential, biologically plausible neuron, single neuron dynamics, computational neuroscience model
metadata	{"arxiv_id":"2607.07429","published":"2023-10-15","authors":"Leon Parepko, Danila Shulepin, Albert Nasybullin","venue":"DCNA 2023 (7th Scientific School Dynamics of Complex Networks)","tags":["spiking-neural-network","neuron-models","computational-neuroscience","mathematical-modeling","biologically-plausible"]}

Single-Entity Spiking Neuron Models: Survey

Overview

Comprehensive survey of mathematical models for biologically plausible single-neuron dynamics. Classifies models by computational complexity, biological fidelity, and use cases. Covers spiking models (integrate-and-fire variants, conductance-based), subthreshold models, and discrete/continuous analogs.

Core Model Taxonomy

1. Integrate-and-Fire (IF) Family

Leaky Integrate-and-Fire (LIF)

• Simplest spiking model: RC circuit with threshold
• τ_m dV/dt = -(V - V_rest) + RI(t), spike when V ≥ V_th
• Computationally efficient, analytically tractable
• Limitation: no adaptation, no subthreshold oscillations

Exponential Integrate-and-Fire (EIF)

• Adds exponential spike initiation: τ_m dV/dt = -(V-V_rest) + Δ_T exp((V-V_T)/Δ_T) + RI
• Captures sharp spike onset (Na+ channel activation)
• Better fit to cortical neuron data than LIF

Adaptive Exponential IF (AdEx)

• EIF + adaptation current w: τ_w dw/dt = a(V-V_rest) - w + bδ(t-t_spike)
• Reproduces adaptation, bursting, regular spiking
• 4-parameter model fits diverse cortical neuron types

Izhikevich Model

• 2D ODE: dv/dt = 0.04v² + 5v + 140 - u + I, du/dt = a(bv - u)
• With reset: v ≥ 30 → v=c, u=u+d
• 4 parameters (a,b,c,d) reproduce 8+ firing patterns
• Best trade-off: biological realism + computational efficiency

2. Conductance-Based Models

Hodgkin-Huxley (HH)

• Gold standard: models Na+, K+, leak conductances
• 4D ODE system with gating variables (m, h, n)
• Biophysically detailed but computationally expensive
• Foundation for all conductance-based models

Morris-Lecar

• 2D reduction of HH: V + one recovery variable w
• Captures excitability types I and II
• Bifurcation analysis reveals saddle-node/Hopf transitions

FitzHugh-Nagumo (FHN)

• 2D simplification: v' = v - v³/3 - w + I, w' = ε(v + a - bw)
• Qualitative excitability, not quantitatively accurate
• Best for theoretical analysis of excitability

3. Discrete/Analog Hybrid Models

Theta Neuron (Ermentrout-Kopell)

• Phase model: dθ/dt = 1 - cos(θ) + (1 + cos(θ))I
• Exact reduction from saddle-node on invariant circle
• Analytically tractable for network analysis

Spike Response Model (SRM)

• Linear superposition of post-synaptic potentials
• V(t) = Σ η(t-t_i) + Σ ε(t-t_i) * I(t)
• Efficient for large-scale network simulation

Key Classification Dimensions

Biological Fidelity vs. Computational Cost

Model	Dimensions	Firing Patterns	Cost
LIF	1	Regular only	Very Low
EIF	1	Regular + sharp onset	Low
AdEx	2	6+ patterns	Low
Izhikevich	2	8+ patterns	Low
Morris-Lecar	2	Type I/II	Medium
FHN	2	Qualitative	Medium
HH	4	Full biophysical	High

Use Case Selection

• Large-scale networks (>10⁴ neurons): LIF, Izhikevich
• Small circuits with adaptation: AdEx, EIF
• Theoretical analysis: FHN, theta neuron
• Biophysical detail: HH, Morris-Lecar
• Hardware implementation: LIF, EIF (simple dynamics)

Membrane Potential Dynamics

Subthreshold Regime

• LIF: exponential decay to rest
• AdEx/EIF: subthreshold resonance possible
• HH: complex ion channel interactions

Spike Initiation

• LIF: instantaneous threshold (unbiological)
• EIF: smooth exponential onset (matches Na+ activation)
• HH: explicit channel dynamics

After-Spike Dynamics

• LIF: hard reset (no refractory period)
• AdEx: adaptation current creates refractory period
• Izhikevich: reset parameters control refractory behavior

Applications

SNN Training

• LIF: surrogate gradient methods (most common)
• AdEx: more realistic but harder to train
• Izhikevich: good for neuromorphic hardware

Neuromorphic Computing

• LIF/EIF: FPGA, Loihi, TrueNorth implementations
• Hardware constraints drive model simplification

Computational Neuroscience

• HH: detailed single-neuron modeling
• AdEx: population-level cortical dynamics
• FHN: theoretical excitability analysis

Pitfalls

• LIF threshold is unbiological: real neurons have smooth spike onset; use EIF/AdEx for realistic onset dynamics
• Izhikevich parameter selection: wrong (a,b,c,d) combinations produce non-physical behavior; validate against target neuron type
• HH stiffness: explicit Euler unstable; use adaptive solvers (RK45, CVODE) for multi-compartment models
• AdEx adaptation timescale: τ_w must match biological data; default values may not fit all neuron types
• Discrete-time simulation: spike timing errors accumulate; use event-driven simulation or small dt (<0.1ms)

References

• Paper: arXiv:2607.07429 (Parepko, Shulepin, Nasybullin, DCNA 2023)
• Izhikevich (2003) "Simple model of spiking neurons" - IEEE TNN
• Brette & Gerstner (2005) "Adaptive exponential integrate-and-fire model" - J Neurophysiol
• Hodgkin & Huxley (1952) "A quantitative description of membrane current" - J Physiol