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Biological neuron modelsA biological 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.
Artificial neuron abstractionThe 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 is the synaptic weight, 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 abstractionIn 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}. IntegrateandfireOne of the earliest models of a neuron was first investigated in 1907 by 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
This model still has a glaring problem in that it has no timedependent memory. If it receives a belowinput signal one day, it will retain that voltage boost forever until it fires again, something that is clearly not reflected in experiment and not consistent in theory. Leaky integrateandfireIn 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]}. HodgkinHuxleyThe 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 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 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. FitzHughNagumoSweeping 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]}. MorrisLecarIn 1981 Morris and Lecar combined HodgkinHuxley and FitzHughNagumo into a voltagegated calcium channel model with a delayedrectifier potassium channel, represented by
where .^{[2]} Expanded neuron modelsWhile 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
The neuron given in standard circuit models can be considered a literal spherical cow, in that it is a simplified model that describes a uniform, spherical neuron. Cable 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 A_{D} = πld 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
The 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 , 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 B_{in,stem}. Then our total neuron conductance is given by
Synaptic transmission
The 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:
As before, is the average 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 conditionsThe 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
Categories: Neuroscience  Biophysics  Computational neuroscience 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Biological_neuron_models". A list of authors is available in Wikipedia. 