# Biological neuron model

A neuron model is a mathematical model of neuron ( a neuron ), which describes the change over time of the membrane potential, or any other characteristic of the cell. These usually differential equations are used. Biophysical basis of such a description is the fact that the voltage having a nerve cell from its surroundings, dynamically changed by so-called ion channels by currents of charged particles, and that these physical processes can be described by the theory of electrical science. Channels, which themselves have a momentum, so for example, are voltage-dependent, can be described that represent the stochastic opening and closing of the channel has its own equations. Together, the equations describing the behavior of the nerve cell, a dynamic system, which is characterized in particular by non-linear equations. These nonlinearities may explain many of the complex behavior of the nerve cells, such as the sudden increase of the membrane potential in an action potential.

## Application

Neuron models are mainly used in computational neuroscience for applications where they are used for the systematic investigation of brain functions. Man trying to replicate the behavior of real neurons, which was measured in electrophysiological experiments in the model and to understand the basis of the equations by one examines them mathematically or simulated in the computer. In this way, predictions for new experiments can be derived. These can then be called to validate the model approach. In the best case, such a model guided approach leads to new experiments and findings on which you would not have come without the modeling.

Neuron models can also be used as a component of neural network models. Here are more interested in the interaction of nerve cells, and can be as similar studies carried out as at the level of individual cells.

Highly idealized neuron models such as the McCulloch - Pitts cell are used in artificial neural networks. In such networks, the highly parallelized, driven by learning processing strategy of the brain is imitated to solve complex technical problems, such as predicting the behavior of a time series or the recognition of patterns in images.

## Types of neuron models

Depending on the application, the different neuron models differed widely in their abstraction of the biophysical conditions.

### Hodgkin -Huxley models

Models of Hodgkin -Huxley - type model explicitly the dynamic behavior of ion channels through their own differential equations. The parameters of these equations are derived directly from electrophysiological measurements. In the original version of the Hodgkin and Huxley model had a sodium and a potassium channel and was described by four differential equations. However, models of this type are also particularly well to the properties of other ion channels (such as calcium channels ) and their impact on the dynamics of the nerve cell to investigate (eg adaptation to repeatedly presented stimuli, refractory period, resonance phenomena ).

### Extensions of the Hodgkin-Huxley models

However, even the Hodgkin -Huxley models are already in various ways idealizations dar. particular ion channels can in fact only be open or closed, and change these states stochastically. Therefore, the modeling of the channels through continuous " gating variable" that have values between zero ( fully closed channel ) and one ( channel fully open) can assume only an approximation is (more precisely, a mean-field approximation ). A more detailed modeling of channel dynamics can be accomplished by using Markov chains, exactly which map such random state transitions.

Secondly, the Hodgkin -Huxley model describes a nerve cell as a point-like structure without geometric expansion (more precisely, as a point process). Thus, the complex morphology of real neurons, particularly their often very large dendritic trees, ignored. Therefore, so-called compartmental models have been developed that treat the individual components ( compartment ) of a nerve cell, such as Soma, axon and dendrites in principle as independent cells. These can then be possibly modeled with different parameters within the Hodgkin -Huxley formalism. The connections between the compartment, ie, the flow of ions inside the cell are, using the cable theory as an electrical circuit treated with branches ( see also: Kirchhoff's rules, network analysis (Electrical ) ). In such models, the experimentally determined morphology real cells can be installed in order to study their effects on the dynamic properties of the cell ( for example, spatial and temporal integration of synaptic stimulation, dendritic spikes). Such models are very often computer simulated with specialized programs such as NEURON.

### Reduced models spike neurons Direction

On the other hand, there were also efforts to simplify the complex Hodgkin -Huxley model, while preserving its essential dynamic properties. These reduced models were introduced, which reduce the number of differential equations or simplify their structure. Examples are the Hindmarsh - Rose model with three differential equations, as well as the Morris - Lecar model, the FitzHugh - Nagumo model and the Izhikevich model with two differential equations. In the two-dimensional models, it is in particular possible to represent the dynamics of the system by using a phase space portrait graphically. This can be in particular the jump in the membrane potential during a spike ( action potential ) mathematically explained as bifurcation and make vividly.

An even greater reduction represents the integrate- and-fire neuron dar. here just is not a passive " leak current " explicitly modeled through the membrane, the generation of the action potential is replaced by an artificial threshold mechanism: Whenever the membrane potential exceeds a threshold value, the potential automatically to a specific value (often the resting potential ) is reset. Thus, the model can depict only processes below the threshold - the summation of synaptic input currents ( integrate), and the " firing" of an action potential when the threshold is reached ( fire ). An invaluable advantage of this model is that you can solve his differential equation explicitly, so that the integrate- and-fire model is often used despite its strong simplification for mathematical analysis of brain functions. In network simulations, it is often used, as it comes with many neurons consumed by its simplicity, even for large networks only little computing time.

### Fire rate models

One approach to further simplify neuron models is no longer considered the membrane potential itself as a dynamic variable, but the rate of fire, so the frequency can be generated with the action potentials, or even more abstract the activation of the cell for no direct reference more physiology has. The neuron is then described as a non-linear transfer function between input rate and output rate, for example, in the form of a sigmoid function. Examples of such models are the continuous basic model and the McCulloch - Pitts cell. Firing rates or activation -based models are used in simulations, which focus on the network structure and the learning of synaptic connections, especially in the field of artificial neural networks.