Neurons and Neural Networks: Computational Models

Abstract

Neural networks produce electrical activity that is generated by the biophysical properties of the constituent neurons and synapses and is affected by several factors including neuromodulators and the network topology. Individual neurons produce electrical signals through processes that are highly nonlinear and communicate these signals to one another through synaptic interactions, resulting in emergent network outputs. The output of neural networks underlies behaviours in all higher animals. Mathematical equations can be used to describe the electrical activity of neurons and neural networks and the underlying biophysical properties. These equations give rise to computational models of neurons and networks that can be analysed using mathematical techniques or numerically simulated with computers.

Key Concepts

  • An action potential is a brief nonlinear rise and fall of the membrane voltage of a neuron and is the primary signal used for neural communication.
  • Excitability is the ability of neurons and other neuron types to produce action potentials when the transmembrane voltage crosses a threshold.
  • The Hodgkin–Huxley model is a mathematical description of how action potentials are generated in neurons and propagate along their axons.
  • The integrate‐and‐fire neuron is a simplified mathematical model of excitability in neurons, useful for the ability to do mathematical analysis and used primarily in network models.
  • Bifurcation is a mathematical term for a change in the qualitative structure of a dynamical system when a parameter value is changed.
  • Neural oscillations are repetitive or rhythmic changes in the voltage activity of a neuron or a network of neurons. Neural oscillations may arise in individual neurons or through network synchrony.
  • Bursting is the ability of some neurons and networks to produce periodic spiking activity followed by an interval of no activity.
  • A half‐centre oscillator is a network of two neurons that produce bursting activity out of phase with one another and are a key subnetwork of central pattern generators.

Keywords: nonlinear dynamics; bifurcation; phase plane; synchrony; membrane biophysics; Hodgkin–Huxley model

Figure 1. Equivalent circuits of neural models. (a) Circuit diagram for a single‐compartment neuron representing ion currents of the Hodgkin–Huxley model. Cm is the membrane capacitance. Resistor symbols indicate ionic conductance (with arrows show varying conductances). Batteries represent the Nernst equilibrium potential for sodium (Na) and potassium (K) and the reversal potential of the passive leak current. (b) The equivalent circuit for a compartmental model, representing a uniform cable such as an axon. Each compartment is represented with the equivalent circuit as in panel a. The circuits representing each compartment are coupled through resistors representing the intracellular resistance (Ri) and the extracellular resistance (Ro). Ellipses indicate that the cable continues in both directions.
Figure 2. Some common models of two‐cell networks. (a) Top: Synaptic inhibition is due to the release of a neurotransmitter that moves the postsynaptic membrane potential away from action potential threshold. Bottom: Two cells that are reciprocally coupled by synaptic inhibition can produce out‐of‐phase oscillatory activity (half‐centre oscillation). (b) Top: Synaptic excitation is caused by a neurotransmitter that moves the postsynaptic membrane potential toward action potential threshold. Bottom: Two cells coupled with reciprocal excitation can oscillate in phase but the action potentials are not necessarily time‐locked. (c) Top: Electrical coupling is due to ion channels (gap junctions) that span the membranes of two cells and allow free flow of ions between the two. Bottom: Electrically coupled cells typically demonstrate synchronous activity, which may be oscillatory even if the two cells are not rhythmically active in isolation.
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Further Reading

Borgers C (2017) An Introduction to Modeling Neuronal Dynamics. Springer.

Dayan P and Abbott LF (2001) Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press, Springer: Cambridge, MA.

Gabbiani F and Cox SJ (2010) Mathematics for Neuroscientists. Elsevier.

Izhikevich E (2006) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press: Cambridge, MA.

Johnston D and Wu SM‐S (1995) Foundations of Cellular Neurophysiology. MIT Press: Cambridge, MA.

Koch C and Segev I (eds) (1998) Methods in Neuronal Modeling: From Ions to Networks, 2nd edn. MIT Press: Cambridge, MA.

Koch C (1999) Biophysics of Computation. Oxford University Press: Oxford, UK.

Miller P (2018) An Introductory Course in Computational Neuroscience. MIT Press: Cambridge, MA.

ModelDB A database to support computational neuroscience. http://senselab.med.yale.edu/senselab/modeldb/.

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Rotstein, Horacio G, and Nadim, Farzan(Mar 2020) Neurons and Neural Networks: Computational Models. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0000089.pub4]