 Biological neuron model

A biological neuron model (also known as spiking neuron model) is a mathematical description of the properties of nerve cells, or neurons, that is designed to accurately describe and predict biological processes. This is in contrast to the artificial neuron, which aims for computational effectiveness, although these goals sometimes overlap.
Contents
Artificial neuron abstraction
The most basic model of a neuron consists of an input with some synaptic weight vector and an activation function or transfer function inside the neuron determining output. This is the basic structure used in artificial neurons, which in a neural network often looks like
where y_{j} is the output of the jth neuron, x_{i} is the ith input neuron signal, w_{ij} is the synaptic weight between the neurons i and j, and φ is the activation function. Some of the earliest biological models took this form until kinetic models such as the HodgkinHuxley model became dominant.
Biological abstraction
In the case of modelling a biological neuron, physical analogues are used in place of abstractions such as "weight" and "transfer function". The input to a neuron is often described by an ion current through the cell membrane that occurs when neurotransmitters cause an activation of ion channels in the cell. We describe this by a physical timedependent current I(t). The cell itself is bound by an insulating cell membrane with a concentration of charged ions on either side that determines a capacitance C_{m}. Finally, a neuron responds to such a signal with a change in voltage, or an electrical potential energy difference between the cell and its surroundings, which is observed to sometimes result in a voltage spike called an action potential. This quantity, then, is the quantity of interest and is given by V_{m}.
Integrateandfire
One of the earliest models of a neuron was first investigated in 1907 by Louis Lapicque^{[1]}. A neuron is represented in time by
which is just the time derivative of the law of capacitance, Q = CV. When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold V_{th}, at which point a delta function spike occurs and the voltage is reset to its resting potential, after which the model continues to run. The firing frequency of the model thus increases linearly without bound as input current increases.
The model can be made more accurate by introducing a refractory period t_{ref} that limits the firing frequency of a neuron by preventing it from firing during that period. Through some calculus involving a Fourier transform, the firing frequency as a function of a constant input current thus looks like
 .
A remaining shortcoming of this model is that it implements no timedependent memory. If the model receives a belowthreshold signal at some time, it will retain that voltage boost forever until it fires again. This characteristic is clearly not in line with observed neuronal behavior.
Leaky integrateandfire
In the leaky integrateandfire model, the memory problem is solved by adding a "leak" term to the membrane potential, reflecting the diffusion of ions that occurs through the membrane when some equilibrium is not reached in the cell. The model looks like
where R_{m} is the membrane resistance, as we find it is not a perfect insulator as assumed previously. This forces the input current to exceed some threshold I_{th} = V_{th} / R_{m} in order to cause the cell to fire, else it will simply leak out any change in potential. The firing frequency thus looks like
which converges for large input currents to the previous leakfree model with refractory period^{[2]}.
HodgkinHuxley
Main article: HodgkinHuxley modelThe most successful and widelyused models of neurons have been based on the Markov kinetic model developed from Hodgkin and Huxley's 1952 work based on data from the squid giant axon. We note as before our voltagecurrent relationship, this time generalized to include multiple voltagedependent currents:
 .
Each current is given by Ohm's Law as
where g(t,V) is the conductance, or inverse resistance, which can be expanded in terms of its constant average ḡ and the activation and inactivation fractions m and h, respectively, that determine how many ions can flow through available membrane channels. This expansion is given by
and our fractions follow the firstorder kinetics
with similar dynamics for h, where we can use either τ and m_{∞} or α and β to define our gate fractions.
With such a form, all that remains is to individually investigate each current one wants to include. Typically, these include inward Ca^{2+} and Na^{+} input currents and several varieties of K^{+} outward currents, including a "leak" current. The end result can be at the small end 20 parameters which one must estimate or measure for an accurate model, and for complex systems of neurons not easily tractable by computer. Careful simplifications of the HodgkinHuxley model are therefore needed.
FitzHughNagumo
Main article: FitzHughNagumo modelSweeping simplifications to HodgkinHuxley were introduced by FitzHugh and Nagumo in 1961 and 1962. Seeking to describe "regenerative selfexcitation" by a nonlinear positivefeedback membrane voltage and recovery by a linear negativefeedback gate voltage, they developed the model described by
where we again have a membranelike voltage and input current with a slower general gate voltage w and experimentallydetermined parameters a = 0.7, b = 0.8, τ = 1/0.08. Although not clearly derivable from biology, the model allows for a simplified, immediately available dynamic, without being a trivial simplification^{[3]}.
MorrisLecar
Main article: MorrisLecar modelIn 1981 Morris and Lecar combined HodgkinHuxley and FitzHughNagumo into a voltagegated calcium channel model with a delayedrectifier potassium channel, represented by
where .^{[2]}
HindmarshRose
Main article: HindmarshRose modelBuilding upon the FitzHughNagumo model, Hindmarsh and Rose proposed in 1984 a model of neuronal activity described by three coupled first order differential equations:
with r^{2} = x^{2} + y^{2} + z^{2}, and r ≈ 10^{2} so that the z variable only changes very slowly. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the x variable of the model, which include chaotic dynamics. This makes the HindmarshRose neuron model very useful, because being still simple, allows a good qualitative description of the many different patterns of the action potential observed in experiments.
Expanded neuron models
While the success of integrating and kinetic models is undisputed, much has to be determined experimentally before accurate predictions can be made. The theory of neuron integration and firing (response to inputs) is therefore expanded by accounting for the nonideal conditions of cell structure.
Cable theory
See also: Cable theoryCable theory describes the dendritic arbor as a cylindrical structure undergoing a regular pattern of bifurcation, like branches in a tree. For a single cylinder or an entire tree, the input conductance at the base (where the tree meets the cell body, or any such boundary) is defined as
 ,
where L is the electrotonic length of the cylinder which depends on its length, diameter, and resistance. A simple recursive algorithm scales linearly with the number of branches and can be used to calculate the effective conductance of the tree. This is given by
where {{mathA_{D} = πld</math> is the total surface area of the tree of total length l, and L_{D} is its total electrotonic length. For an entire neuron in which the cell body conductance is G_{S} and the membrane conductance per unit area is G_{md} = G_{m} / A, we find the total neuron conductance G_{N} for n dendrite trees by adding up all tree and soma conductances, given by
 ,
where we can find the general correction factor F_{dga} experimentally by noting G_{D} = G_{md}A_{D}F_{dga}.
Compartmental models
See also: Multicompartment modelThe cable model makes a number of simplifications to give closed analytic results, namely that the dendritic arbor must branch in diminishing pairs in a fixed pattern. A compartmental model allows for any desired tree topology with arbitrary branches and lengths, but makes simplifications in the interactions between branches to compensate. Thus, the two models give complementary results, neither of which is necessarily more accurate.
Each individual piece, or compartment, of a dendrite is modeled by a straight cylinder of arbitrary length l and diameter d which connects with fixed resistance to any number of branching cylinders. We define the conductance ratio of the ith cylinder as B_{i} = G_{i} / G_{∞}, where and R_{i} is the resistance between the current compartment and the next. We obtain a series of equations for conductance ratios in and out of a compartment by making corrections to the normal dynamic B_{out,i} = B_{in,i+1}_{, as}
 _{}
 _{}
 _{}
_{where the last equation deals with parents and daughters at branches, and . We can iterate these equations through the tree until we get the point where the dendrites connect to the cell body (soma), where the conductance ratio is Bin,stem. Then our total neuron conductance is given by}
 .
Synaptic transmission
See also: NeurotransmissionThe response of a neuron to individual neurotransmitters can be modeled as an extension of the classical HodgkinHuxley model with both standard and nonstandard kinetic currents. Four neurotransmitters have primarily influence in the CNS. AMPA/kainate receptors are fast excitatory mediators while NMDA receptors mediate considerably slower currents. Fast inhibitory currents go through GABA_{A} receptors, while GABA_{B} receptors mediate by secondary Gproteinactivated potassium channels. This range of mediation produces the following current dynamics:
where ḡ is the maximal^{[4]}^{[5]} conductance (around 1S) and E is the equilibrium potential of the given ion or transmitter (AMDA, NMDA, Cl, or K), while [O] describes the fraction of receptors that are open. For NMDA, there is a significant effect of magnesium block that depends sigmoidally on the concentration of intracellular magnesium by B(V). For GABA_{B}, [G] is the concentration of the Gprotein, and K_{d} describes the dissociation of G in binding to the potassium gates.
The dynamics of this more complicated model have been wellstudied experimentally and produce important results in terms of very quick synaptic potentiation and depression, that is, fast, shortterm learning.
Other conditions
The models above are still idealizations. Corrections must be made for the increased membrane surface area given by numerous dendritic spines, temperatures significantly hotter than roomtemperature experimental data, and nonuniformity in the cell's internal structure^{[2]}. Many problems in the temperature and geometry dynamics of the cell during action potential propagation, as well as problems in explaining some pharmacology, are still unsolved, some of which have required unorthodox new models, such as the soliton model, to explain.
References
 ^ Abbott, L.F. (1999). "Lapique's introduction of the integrateandfire model neuron (1907)". Brain Research Bulletin 50 (5/6): 303–304. doi:10.1016/S03619230(99)001616. PMID 10643408. http://neurotheory.columbia.edu/~larry/AbbottBrResBul99.pdf. Retrieved 20071124.
 ^ ^{a} ^{b} ^{c} Koch, Christof; Idan Segev (1998). Methods in Neuronal Modeling (2 ed.). Cambridge, MA: Massachusetts Institute of Technology. ISBN 0262112310.
 ^ Izhikevich, Eugene M.; Richard FitzHugh. "FitzHughNagumo Model". Scholarpedia. http://www.scholarpedia.org/article/FitzHughNagumo_Model. Retrieved 20071125.
 ^ Hodgkin & Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, August 1952, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1392413/
 ^ Christof Koch & Idan Segev, Methods in Neuronal Modeling, 1989, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.9768&rep=rep1&type=pdf
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
Neuron — This article is about cells in the nervous system. For other uses, see Neuron (disambiguation). Brain cell redirects here. For other uses, see Glial cell. Neuron: Nerve Cell … Wikipedia
Morris–Lecar model — The Morris–Lecar model is a biological neuron model developed by Catherine Morris and Harold Lecar to reproduce the variety of oscillatory behavior in relation to Ca++ and K+ conductance in the giant barnacle muscle fiber.[1] Morris Lecar neurons … Wikipedia
Hodgkin–Huxley model — Basic components of Hodgkin–Huxley type models. Hodgkin–Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (Cm). Voltage gated and leak ion channels are represented by… … Wikipedia
Artificial neuron — An artificial neuron is a mathematical function conceived as a crude model, or abstraction of biological neurons. Artificial neurons are the constitutive units in an artificial neural network. Depending on the specific model used, it can receive… … Wikipedia
FitzHugh–Nagumo model — The FitzHugh Nagumo model (named after Richard FitzHugh, 1922 ndash;2007) describes a prototype of an excitable system, i.e., a neuron.If the external stimulus I { m ext} exceeds a certain threshold value, the system will exhibit a characteristic … Wikipedia
Soliton model — The Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons. It proposes that the signals travel along the cell s membrane in the form of certain kinds of sound (or density)… … Wikipedia
HindmarshRose model — The Hindmarsh Rose model of neuronal activity is aimed to study the spiking bursting behavior of the membrane potential observed in experiments made with a single neuron. The relevant variable is the membrane potential, x ( t ), which is written… … Wikipedia
Multicompartment model — A multi compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the compartments of a system. Each compartment is assumed to be a homogenous entity within which the entities being… … Wikipedia
Mirror neuron — A mirror neuron is a neuron that fires both when an animal acts and when the animal observes the same action performed by another.[1][2] [3] Thus, the neuron mirrors the behaviour of the other, as though the observer were itself acting. Such… … Wikipedia
Modelling biological systems — Modeling biological systems is a significant task of systems biology and mathematical biology. Computational systems biology aims to develop and use efficient algorithms, data structures, visualization and communication tools with the goal of… … Wikipedia