## 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. 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. This article briefly reviews the current mathematical and computational techniques involved in modelling neurons and neural networks.

#### Key Concepts:

• An action potential is a brief nonlinear rise and fall of the membrane voltage of a cell and is the primary signal used for neural communication.

• Excitability is the ability of neurons and other cell 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, is useful for the ability to do mathematical analysis and is 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 probursting activity out of phase with one another and is a key subnetwork of central pattern generators.

Keywords: bifurcation; phase plane; synchrony; Hodgkin–Huxley model; compartmental modelling; cable equation; balanced networks

 Figure 1. Equivalent circuits of neural models. (a) Circuit diagram for a single‐compartment neuron representing ionic currents of the Hodgkin–Huxley model. Cm is the membrane capacitance. Resistor symbols indicate ionic conductance (with arrows show varying conductance). 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 towards 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.

References

Bower JM and Beeman D (1998) The book of GENESIS: Exploring realistic neural models with the General NEural SImulation System. Cambridge, UK: Cambridge University Press.

Brown TG (1914) On the nature of the fundamental activity of the nervous centres; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. Journal of Physiology 48: 18–46.

Carnevale NT and Hines ML (2006) The NEURON book. Cambridge, UK: Cambridge University Press.

Catterall WA , Raman IM , Robinson HP , Sejnowski TJ and Paulsen O (2012) The Hodgkin–Huxley heritage: from channels to circuits. Journal of Neuroscience 32: 14064–14073.

Dayan P and Abbott LF (2001) Theoretical neuroscience: Computational and mathematical modeling of neural systems. Cambridge, MA: MIT Press.

Ermentrout B (2002) Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, 1st edn. Philadelphia, PA: Soc for Industrial & Applied Math.

Ermentrout B and Terman DH (2010) Mathematical Foundations of Neuroscience. New York: Springer.

Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal 1: 445–466.

Goodman DFM and Brette R (2008) The brian simulator. Frontiers in Neuroscience 3. Doi: 10.3389/neuro.01.026.2009

Harris KD , Csicsvari J , Hirase H , Dragoi G and Buzsaki G (2003) Organization of cell assemblies in the hippocampus. Nature 424: 552–556.

Hodgkin AL and Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 117: 500–544.

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

Izhikevich EM (2000) Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos 10: 1171–1266.

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

Kopell N and Ermentrout GB (2000) Mechanisms of phase‐locking and frequency control in pairs of coupled neural oscillators. In: Fiedler B , Iooss G and Kopell N (eds) Handbook of Dynamical Systems, vol. 2, pp 3–54. The Netherlands: Elsevier.

Lapicque L (1907) Recherches quantitatives sur l'excitation electrique des nerfs traitee comme une polarization. Journal de Physiologie et Pathologie Général 9: 620–635.

Marder E (2002) Non‐mammalian models for studying neural development and function. Nature 417: 318–321.

McCulloch WS and Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biology 5: 115–133.

Nadim F , Manor Y , Nusbaum MP and Marder E (1998) Frequency regulation of a slow rhythm by a fast periodic input. Journal of Neuroscience 18: 5053–5067.

Nagumo J and Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proceedings of the IRE 50: 2061–2070.

Prinz AA , Billimoria CP and Marder E (2003) Alternative to hand‐tuning conductance‐based models: construction and analysis of databases of model neurons. Journal of Neurophysiology 90: 3998–4015.

Rall W , Segev I , Rinzel J and Shepherd GM (1995) The theoretical foundation of dendritic function: selected papers of Wilfrid Rall with commentaries. Cambridge, MA: MIT Press.

Rinzel J (1987) A formal classification of bursting mechanisms in exciatable systems. Berlin: Springer‐Verlag.

Rinzel J and Ermentrout GB (1998) Analysis of neural excitability and oscillations. In: Koch C and Segev I (eds) Methods in Neuronal Modeling: From Ions to Networks, pp 251–291. Cambridge, MA: MIT Press.

Vogels TP , Rajan K and Abbott LF (2005) Neural network dynamics. Annual Review of Neuroscience 28: 357–376.

Worgotter F and Porr B (2005) Temporal sequence learning, prediction, and control: a review of different models and their relation to biological mechanisms. Neural Computation 17: 245–319.

Gabbiani F and Cox SJ (2010). Mathematics for Neuroscientists. London, UK: Academic Press.

Koch C (1999) Biophysics of Computation. New York, NY: Oxford University Press.

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

ModelDB: A Database to Support Computational Neuroscience: http://senselab.med.yale.edu/senselab/modeldb/

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