# Computational neuroscience Computational neuroscience explains the [[Nervous_system|nervous system]] with mathematics: it builds [[Mathematical_model|models]] — from single ion channels to whole-brain networks — that make the brain's electrical and chemical activity computable, testable, and comparable to theory. Where experimental [[Neuroscience|neuroscience]] records what neurons do, the computational branch asks what they are *for*: what code the spikes carry ([[Information_theory|information theory]]), what dynamics the circuits implement ([[Dynamical_system|dynamical systems]]), and what problems the whole apparatus solves ([[Control_theory|control]], inference, learning). The field is a direct heir of [[Cybernetics|cybernetics]], and its house style is the physicist's: differential equations, [[Statistical_mechanics|statistical mechanics]], and honest fits to data. ## From cybernetic neurons to a named field The founding artifact is the [[Warren_Sturgis_McCulloch|McCulloch]]–[[Walter_Pitts|Pitts]] neuron (1943): threshold [[Logic|logic]] units whose networks can compute anything a [[Finite-state_machine|finite-state machine]] can — a result that fed straight into [[John_von_Neumann|von Neumann's]] computer architecture and [[Norbert_Wiener|Wiener's]] [[Cybernetics|cybernetics]]. The biophysical anchor came in 1952: Hodgkin and Huxley fit voltage-clamp data from the squid giant axon with four coupled [[Ordinary_differential_equation|ordinary differential equations]] and *predicted* the action potential's shape and ~20 m/s conduction speed (Nobel Prize, 1963). Hebb's 1949 postulate — synapses strengthen when pre- and postsynaptic activity coincide — supplied the learning principle; Rall's cable theory (1950s–60s) put dendrites on mathematical footing; David Marr's three levels (computation, algorithm, implementation; *Vision*, 1982) gave the field its methodological creed; and the name itself was fixed by Eric Schwartz's 1985 conference and the volume that followed. ## The biophysics: neurons as nonlinear circuits A neuron is an excitable membrane: a capacitor holding roughly −70 mV of resting [[Voltage|voltage]], studded with [[Ion|ion]] channels whose conductances depend on that same voltage — the [[Nonlinear_system|nonlinearity]] that makes everything interesting. Depolarize past threshold and sodium influx feeds itself in a [[Positive_feedback|positive-feedback]] spike (~100 mV excursion, ~1 ms), terminated by potassium's slower [[Negative_feedback|negative feedback]]. The Hodgkin–Huxley equations capture this exactly but expensively, so the field maintains a ladder of reductions: FitzHugh–Nagumo (1961) keeps the phase-plane geometry; integrate-and-fire (Lapicque, 1907) keeps only threshold and reset; Izhikevich's 2003 model buys most spiking phenomenology for two equations. Simulators such as NEURON make the [[Simulation|simulation]] of morphologically detailed cells routine, and the choice of rung is a modeling decision about which [[Electric_current|currents]] matter for the question asked. ## Dynamics: attractors, oscillations, criticality Circuit-level theory is applied [[Dynamical_system|dynamical systems]]. Persistent activity is modeled as an [[Attractor|attractor]]: Hopfield's 1982 network made memories fixed points of an energy function borrowed from spin physics, and ring attractors posited for head-direction cells were later observed almost literally in the fly's central complex. [[Bifurcation_theory|Bifurcation]] analysis classifies how neurons switch from rest to rhythmic firing; coupled-oscillator theory explains brain rhythms and their [[Synchronization|synchronization]] across regions (gamma ≈ 30–80 Hz), with [[Oscillation|oscillations]] hypothesized to gate communication between areas. [[Multistability|Multistability]] models perceptual rivalry. At population scale, neuronal avalanches with power-law size statistics (Beggs and Plenz, 2003) suggested cortex may sit near a [[Phase_transition|phase transition]] — the [[Self-organized_criticality|self-organized criticality]] hypothesis, still productively contested — and mean-field methods from [[Statistical_mechanics|statistical mechanics]] underwrite the population equations (Wilson–Cowan, 1972) used to model whole regions. ## Codes: what spikes say The coding wing treats the brain as a communication and inference machine. [[Claude_Shannon|Shannon's]] [[Information_theory|information theory]] quantifies what a spike train tells a downstream reader: [[Entropy_(information_theory)|entropy]] bounds capacity, mutual information measures what stimulus features survive the [[Communication_channel|channel]], and measured rates run from below 1 to a few hundred bits per second per neuron depending on system. Barlow's efficient-coding hypothesis (1961) — sensory neurons should decorrelate natural inputs — predicted receptive-field structure later confirmed in retina and cortex. Perception is modeled as statistical inference: [[Detection_theory|signal detection theory]] for thresholds, Bayesian [[Estimation_theory|estimation]] for cue combination (humans weight cues near-optimally by reliability), [[Kalman_filter|Kalman filters]] for sensorimotor [[Feedback|feedback]] control of movement, and predictive-coding schemes in which cortex trades [[Probability_distribution|probabilistic]] predictions against errors. The productive tension is between such normative theories and what circuits demonstrably implement. ## Learning: synapses, dopamine, and deep networks Plasticity gives the models their memory. Spike-timing-dependent plasticity (Bi and Poo, 1998) sharpened Hebb: potentiation when presynaptic spikes lead postsynaptic ones by ~10–20 ms, depression when they lag. The celebrated bridge to behavior is [[Reinforcement_learning|reinforcement learning]]: midbrain dopamine neurons signal reward-prediction error almost exactly as temporal-difference algorithms require (Schultz, Dayan, Montague, 1997), tying a neuromodulator to a named [[Algorithm|algorithm]]. Traffic with [[Machine_learning|machine learning]] now runs both directions — [[Connectionism|connectionist]] ideas became deep [[Neural_network_(machine_learning)|neural networks]], and those networks returned as the current best predictive models of ventral-stream visual responses — while [[Artificial_intelligence|AI]] and [[Cognitive_science|cognitive science]] mine the brain for architectures, and theorists worry publicly about whether backpropagation has any biologically plausible twin. ## Scale: connectomes and whole-brain models The integrative frontier is structural and grand. A human brain carries ~86 billion neurons (Herculano-Houzel, 2009) and on the order of 10¹⁴ synapses; [[Connectomics|connectomics]] has mapped complete wiring for *C. elegans* (302 neurons; White et al., 1986) and, since 2023–24, an entire adult fly brain of ~140,000 neurons. [[Graph_theory|Graph-theoretic]] analysis of such networks — and of human [[Magnetic_resonance_imaging|MRI]] connectivity inferred via [[Hemoglobin|hemoglobin's]] BOLD signal — finds [[Small-world_network|small-world]] topology, heavy-tailed degree distributions, and rich-club hubs whose [[Centrality|centrality]] marks vulnerability in disease, the working territory of [[Network_science|network science]] and [[Systems_neuroscience|systems neuroscience]]. Whole-brain simulation projects (Blue Brain, 2005–; the EU Human Brain Project, 2013–23) tested how far brute [[Simulation|simulation]] scales; the sober consensus is that models earn their size only when a [[Neural_circuit|circuit-level]] hypothesis, not a supercomputer budget, sets the resolution. **On the spine:** [[Neuroscience]] · [[Dynamical_system]] · [[Information_theory]] · [[Neural_network_(machine_learning)]] · [[Connectomics]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Computational_neuroscience) : [Wikitube](https://en.wikitube.io/wiki/Computational_neuroscience) ## Previous hub tags Hubs: `Systems`. Portals: [[PORTAL_Systems]], [[PORTAL_Graph_theory]], [[PORTAL_Cybernetics]], [[PORTAL_Decision_theory]], [[PORTAL_Information_theory]]. --- *Repopulated 2026-08-12 · redlink fill · 0 deletions.*