Biological neuron model

Biological neuron models, also known as spiking 'neuron models', are mathematical descriptions of the conduction of electrical signals in neurons. Neurons are electrically excitable cells within the nervous system, able to fire electric signals, called action potentials, across a neural network. These mathematical models describe the role of the biophysical and geometrical characteristics of neurons on the conduction of electrical activity.
Central to these models is the description of how the membrane potential across the cell membrane changes over time. In an experimental setting, stimulating neurons with an electrical current generates an action potential, that propagates down the neuron's axon. This axon can branch out and connect to a large number of downstream neurons at sites called synapses. At these synapses, the spike can cause the release of neurotransmitters, which in turn can change the voltage potential of downstream neurons. This change can potentially lead to even more spikes in those downstream neurons, thus passing down the signal. As many as 95% of neurons in the neocortex, the outermost layer of the mammalian brain, consist of excitatory pyramidal neurons, and each pyramidal neuron receives tens of thousands of inputs from other neurons. Thus, spiking neurons are a major information processing unit of the nervous system.
One such example of a spiking neuron model may be a highly detailed mathematical model that includes spatial morphology. Another may be a conductance-based neuron model that views neurons as points and describes the membrane voltage dynamics as a function of trans-membrane currents. A mathematically simpler "integrate-and-fire" model significantly simplifies the description of ion channel and membrane potential dynamics.

Biological background, classification, and aims of neuron models

Non-spiking cells, spiking cells, and their measurement
Not all the cells of the nervous system produce the type of spike that defines the scope of the spiking neuron models. For example, cochlear hair cells, retinal receptor cells, and retinal bipolar cells do not spike. Furthermore, many cells in the nervous system are not classified as neurons but instead are classified as glia.
Neuronal activity can be measured with different experimental techniques, such as the "Whole cell" measurement technique, which captures the spiking activity of a single neuron and produces full amplitude action potentials.
With extracellular measurement techniques, one or more electrodes are placed in the extracellular space. Spikes, often from several spiking sources, depending on the size of the electrode and its proximity to the sources, can be identified with signal processing techniques. Extracellular measurement has several advantages:

It is easier to obtain experimentally;
It is robust and lasts for a longer time;
It can reflect the dominant effect, especially when conducted in an anatomical region with many similar cells.

Overview of neuron models
Neuron models can be divided into two categories according to the physical units of the interface of the model. Each category could be further divided according to the abstraction/detail level:

[|Electrical input–output membrane voltage models] – These models produce a prediction for membrane output voltage as a function of electrical stimulation given as current or voltage input. The various models in this category differ in the exact functional relationship between the input current and the output voltage and in the level of detail. Some models in this category predict only the moment of occurrence of the output spike ; other models are more detailed and account for sub-cellular processes. The models in this category can be either deterministic or probabilistic.
[|Natural] stimulus or [|pharmacological input neuron models] – The models in this category connect the input stimulus, which can be either pharmacological or natural, to the probability of a spike event. The input stage of these models is not electrical but rather has either pharmacological concentration units, or physical units that characterize an external stimulus such as light, sound, or other forms of physical pressure. Furthermore, the output stage represents the probability of a spike event and not an electrical voltage.

Although it is not unusual in science and engineering to have several descriptive models for different abstraction/detail levels, the number of different, sometimes contradicting, biological neuron models is exceptionally high. This situation is partly the result of the many different experimental settings, and the difficulty to separate the intrinsic properties of a single neuron from measurement effects and interactions of many cells.
Aims of neuron models
Ultimately, biological neuron models aim to explain the mechanisms underlying the operation of the nervous system. However, several approaches can be distinguished, from more realistic models to more pragmatic models. Modeling helps to analyze experimental data and address questions. Models are also important in the context of restoring lost brain functionality through neuroprosthetic devices.

Electrical input–output membrane voltage models

The models in this category describe the relationship between neuronal membrane currents at the input stage and membrane voltage at the output stage. This category includes integrate-and-fire models and biophysical models inspired by the work of Hodgkin–Huxley in the early 1950s using an experimental setup that punctured the cell membrane and allowed to force a specific membrane voltage/current.
Most modern [|electrical neural interfaces] apply extra-cellular electrical stimulation to avoid membrane puncturing, which can lead to cell death and tissue damage. Hence, it is not clear to what extent the electrical neuron models hold for extra-cellular stimulation.

Hodgkin–Huxley

The Hodgkin–Huxley model
is a model of the relationship between the flow of ionic currents across the neuronal cell membrane and the membrane voltage of the cell. It consists of a set of nonlinear differential equations describing the behavior of ion channels that permeate the cell membrane of the squid giant axon. Hodgkin and Huxley were awarded the 1963 Nobel Prize in Physiology or Medicine for this work.
It is important to note the voltage-current relationship, with multiple voltage-dependent currents charging the cell membrane of capacity
The above equation is the time derivative of the law of capacitance, where the change of the total charge must be explained as the sum over the currents. Each current is given by
where is the conductance, or inverse resistance, which can be expanded in terms of its maximal conductance and the activation and inactivation fractions and, respectively, that determine how many ions can flow through available membrane channels. This expansion is given by
and our fractions follow the first-order kinetics
with similar dynamics for, where we can use either and or and to define our gate fractions.
The Hodgkin–Huxley model may be extended to include additional ionic currents. Typically, these include inward Ca²⁺ and Na⁺ input currents, as well as several varieties of K⁺ outward currents, including a "leak" current.
The result can be at the small end of 20 parameters which one must estimate or measure for an accurate model. In a model of a complex system of neurons, numerical integration of the equations are computationally expensive. Careful simplifications of the Hodgkin–Huxley model are therefore needed.
The model can be reduced to two dimensions thanks to the dynamic relations which can be established between the gating variables. it is also possible to extend it to take into account the evolution of the concentrations.

Perfect Integrate-and-fire

One of the earliest models of a neuron is the perfect integrate-and-fire model, first investigated in 1907 by Louis Lapicque. A neuron is represented by its membrane voltage which evolves in time during stimulation with an input current according
which is just the time derivative of the law of capacitance,. When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold, 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 that limits the firing frequency of a neuron by preventing it from firing during that period. For constant input the threshold voltage is reached after an integration time after starting from zero. After a reset, the refractory period introduces a dead time so that the total time until the next firing is . The firing frequency is the inverse of the total inter-spike interval. The firing frequency as a function of a constant input current, is therefore
A shortcoming of this model is that it describes neither adaptation nor leakage. If the model receives a below-threshold short current pulse at some time, it will retain that voltage boost forever - until another input later makes it fire. This characteristic is not in line with observed neuronal behavior. The following extensions make the integrate-and-fire model more plausible from a biological point of view.

Leaky integrate-and-fire

The leaky integrate-and-fire model, which can be traced back to Louis Lapicque, contains a "leak" term in the membrane potential equation that reflects the diffusion of ions through the membrane, unlike the non-leaky integrate-and-fire model. The model equation looks like
where is the voltage across the cell membrane and is the membrane resistance.. The model equation is valid for arbitrary time-dependent input until a threshold is reached; thereafter the membrane potential is reset.
For constant input, the minimum input to reach the threshold is. Assuming a reset to zero, the firing frequency thus looks like
which converges for large input currents to the previous leak-free model with the refractory period. The model can also be used for inhibitory neurons.
The most significant disadvantage of this model is that it does not contain neuronal adaptation, so that it cannot describe an experimentally measured spike train in response to constant input current. This disadvantage is removed in generalized integrate-and-fire models that also contain one or several adaptation-variables and are able to predict spike times of cortical neurons under current injection to a high degree of accuracy.