Neuron model

from Wikipedia, the free encyclopedia

A neuron model is a mathematical model of a nerve cell (a neuron) that describes the change in membrane potential or another characteristic of the cell over time.

Differential equations are mostly used for this. The biophysical basis of such a description is the fact that the voltage that a nerve cell exhibits in relation to its environment changes dynamically through the currents of charged particles through so-called ion channels , and that these physical processes can be described by the theory of electricity . Channels that have dynamics themselves, for example, are voltage-dependent, can be described using their own equations that map the stochastic opening and closing of the channel.

Together, the equations that describe the behavior of the nerve cell form a dynamic system that is characterized in particular by non-linear equations. These nonlinearities can explain many of the complex behaviors of nerve cells, for example the sudden increase in membrane potential at an action potential .

application

Neuron models are mainly used in computational neuroscience , where they are used for the systematic investigation of brain functions . Attempts are made to model the behavior of real nerve cells, which were measured in electrophysiological experiments, and to understand them using the equations by examining them mathematically or simulating them on the computer . In this way, predictions for new experiments can also be derived. These can then be used to check the model. In the best case, such a model-guided approach leads to new experiments and findings that would not have been made without the modeling.

Neuron models are also used as part of neural network models. Here one is interested in the interaction of nerve cells and can thus carry out studies similar to those on the level of the individual cells.

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

Types of neuron models

Depending on the area of ​​application, the different neuron models differ greatly in their abstraction from the biophysical conditions.

Hodgkin-Huxley models

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

Extensions to the Hodgkin-Huxley models

However, even the Hodgkin-Huxley models represent idealizations in various respects . In particular, ion channels can in reality only be open or closed and change these states stochastically. Therefore, the modeling of the channels using continuous "gating variables", which can assume values ​​between zero (channel completely closed) and one (channel completely open), only represents an approximation (more precisely: a mean field approximation). A more precise modeling of the channel dynamics can be done with the help of Markov chains , which map precisely such random state transitions.

On the other hand, the Hodgkin-Huxley model describes a nerve cell as a point-like structure without geometric expansion (more precisely: as a stochastic point process ). This ignores the complex morphology of real nerve cells, in particular their often widely branched dendrite trees. For this reason, so-called compartment models were developed, which in principle treat the individual components ( compartments ) of a nerve cell such as soma , axon and dendrites like independent cells. These can then be modeled, possibly with different parameters, within the framework of the Hodgkin-Huxley formalism. The connections between the compartments, i.e. the flow of ions within the cell, are treated like an electrical circuit with branches using cable theory (see also: Kirchhoff's rules , network analysis (electrical engineering) ). The experimentally determined morphology of real cells can be incorporated into such models in order to study their effects on the dynamic properties of the cell (e.g. spatial and temporal integration of synaptic stimuli, dendritic spikes). Such models are very often simulated in the computer with specialized programs such as NEURON .

Reduced models of spiking neurons

On the other hand, there were also efforts to simplify the complex Hodgkin-Huxley model, while retaining its essential dynamic properties. For this purpose, reduced models were introduced that 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 particularly possible to graphically depict the dynamics of the system using a phase space portrait. In particular, the sudden increase in membrane potential in the event of a spike (action potential) can be mathematically explained as a bifurcation and made clear.

The integrate-and-fire neuron represents an even greater reduction . Here, only a passive "leakage current" through the membrane is explicitly modeled, the generation of the action potential is replaced by an artificial threshold value mechanism: Whenever the membrane potential exceeds a threshold value, the potential is automatically reset to a certain value (often the resting potential ). This means that the model can only depict processes below the threshold value - the summing up of synaptic input currents ( integrate ) and the "firing" of an action potential when the threshold value is reached ( fire ). An invaluable advantage of this model is that you can solve your differential equation explicitly, so that the Integrate-and-Fire model, despite its great simplification, is often used for mathematical analyzes of brain functions. It is also used very often in network simulations because its simplicity means that it requires little computing time even in large networks with many neurons.

Rate of fire models

One approach to further simplify neuron models is to no longer regard the membrane potential itself as a dynamic variable, but rather the rate of fire , i.e. the frequency with which action potentials are generated, or, more abstractly, the activation of the cell, which no longer has a direct relationship to physiology Has. The neuron is then described as a non-linear transfer function between input rate and output rate, e.g. B. in the form of a sigmoid function . Examples of such models are the continuous basic model and the McCulloch-Pitts cell. Fire rate or activation-based models are used in simulations that focus on the network structure and the learning of synaptic connections, especially in the field of artificial neural networks.

See also

literature

  • Eugene M. Izhikevich: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting . MIT Press, Cambridge, Mass 2007, ISBN 0-262-09043-0 .
  • Larry F. Abbott, Peter Dayan: Theoretical neuroscience: computational and mathematical modeling of neural systems . MIT Press, Cambridge, Mass 2001, ISBN 0-262-04199-5 .
  • William Bialek, Fred Rieke, David Warland, Rob de Ruyter van Steveninck: Spikes: exploring the neural code . MIT Press, Cambridge, Mass 1999, ISBN 0-262-68108-0 .

Web links