One neuron as an equation
A neuron keeps a voltage across its membrane. Inputs from other neurons push that voltage up or down, and when it crosses a threshold the neuron fires a spike. In 1952, Alan Hodgkin and Andrew Huxley described this process in the squid giant axon with a set of differential equations for how sodium and potassium currents change the voltage. The work earned a Nobel Prize in 1963 and remains the foundation of the field.
C dV/dt = −gNam³h(V − ENa) − gKn⁴(V − EK) − gL(V − EL) + I
Simpler models for bigger networks
Hodgkin–Huxley is detailed but expensive to simulate in large numbers. The leaky integrate-and-fire neuron, whose roots go back to Louis Lapicque in 1907, keeps the essentials: the voltage drifts back to rest, inputs add up, and a spike fires at threshold before the voltage resets.
τ dV/dt = −(V − Vrest) + R·I(t); if V ≥ Vth: spike, V → Vreset
Simpler still are rate models, which track a neuron's average firing rate instead of individual spikes. Artificial neural networks used in modern AI are distant descendants of these.
From neurons to networks
Connect many model neurons with a weight matrix W, the mathematical form of a connectome, and you get a network whose state changes over time:
xt+1 = f(W xt + U st + b)
Here x is the activity of every neuron, s is sensory input and f is a nonlinear function. When W contains loops, activity can echo, sustain itself and settle into patterns. These recurrent networks can hold memories, make decisions over time and generate rhythms.
Brains as dynamical systems
A powerful way to think about networks is as dynamical systems moving through a high-dimensional state space. Stable states called attractors can represent memories, as in John Hopfield's 1982 network. Trajectories can represent movements or thoughts unfolding in time. Many neuroscientists now analyse recordings of thousands of neurons this way, looking for the low-dimensional structure hidden in the activity.
Why mathematics matters
Equations force precision. A model makes predictions that can fail, can be tested against recordings, and can be changed one piece at a time to see what each piece does. That makes computational neuroscience the natural bridge between what brains do and what we might build.
We use these tools to build our artificial systems. Starting from the network equation above, we add a persistent self-model and a recurrent workspace, then lesion them selectively to see which mechanisms change behaviour. Mathematics first, then the mind.
Quick answers
What is computational neuroscience?
It is the field that uses mathematical models, theory and computer simulation to explain how nervous systems process information, from single neurons to whole networks.
What is the Hodgkin–Huxley model?
A 1952 set of differential equations describing how ion currents through a neuron's membrane generate electrical spikes. It earned Hodgkin and Huxley the 1963 Nobel Prize in Physiology or Medicine.
Further reading
- Hodgkin, A. & Huxley, A. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117.
- Hopfield, J. (1982). Neural networks and physical systems with emergent collective computational abilities. PNAS, 79.
- Dayan, P. & Abbott, L. (2001). Theoretical Neuroscience. MIT Press.
- Gerstner, W., Kistler, W., Naud, R. & Paninski, L. (2014). Neuronal Dynamics. Cambridge University Press.