Biological neuron model


A biological neuron model, also known as a spiking neuron model, is a mathematical description of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecond in duration, as shown in Fig. 1. Spiking neurons are known to be a major signaling unit of the nervous system, and for this reason characterizing their operation is of great importance. It is worth noting that not all the cells of the nervous system produce the type of spike that define 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.
Ultimately, biological neuron models aim to explain the mechanisms underlying the operation of the nervous system for the purpose of restoring lost control capabilities such as perception, decision making, and continuous limb control. In that sense, biological neuron models differ from artificial neuron models that do not presume to predict the outcomes of experiments involving the biological neural tissue. Accordingly, an important aspect of biological neuron models is experimental validation, and the use of physical units to describe the experimental procedure associated with the model predictions.
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:
  1. [|Electrical input–output membrane voltage models] – These models produce a prediction for membrane output voltage as function of electrical stimulation at the input stage. 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 details. Some models in this category are black box models and distinguish only between two measured voltage levels: the presence of a spike or a quiescent state. Other models are more detailed and account for sub-cellular processes.
  2. [|Natural] or [|pharmacological input neuron models] – The models in this category connect between 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. Typically, this output probability is normalized a time constant, and the resulting normalized probability is called the "firing rate" and has units of Hertz. The probabilistic description taken by the models in this category was inspired from laboratory experiments involving either natural or pharmacological stimulation which exhibit variability in the resulting spike pattern. Nevertheless, when averaging these experimental results across several trials, a clear pattern is often revealed.
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 measurements effects and interactions of many cells. To accelerate the convergence to a unified theory, we list several models in each category, and where applicable, also references to supporting experiments.

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. The most extensive experimental inquiry in this category of models was made by 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.

Integrate-and-fire

One of the earliest models of a neuron was first investigated in 1907 by Louis Lapicque. A neuron is represented in time by
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. 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 time-dependent memory. If the model receives a below-threshold 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.

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 behaviour 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.
We note as before our voltage-current relationship, this time generalized to include multiple voltage-dependent currents:
Each current is given by Ohm's law as
where is the conductance, or inverse resistance, which can be expanded in terms of its constant average 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 expanded 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 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 Hodgkin–Huxley model are therefore needed.

Leaky integrate-and-fire

In the leaky integrate-and-fire model, also studied by Louis Lapicque, 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 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 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 leak-free model with refractory period. The model can also be extended to inhibitory neurons.

Fractional-order leaky integrate-and-fire

Recent advances in computational and theoretical fractional calculus lead to a new form of model, called Fractional-order leaky integrate-and-fire developed by Teka et al. The great advantage of this model is that it can capture and integrate all the past activities and can reproduce the time dependent spiking adaptations observed on pyramidal neurons. The model has the following form - more details can be found in.

Galves–Löcherbach

The Galves–Löcherbach model is a specific development of the leaky integrate-and-fire model. It is inherently stochastic. It was developed by mathematicians Antonio Galves and Eva Löcherbach.
Given the model specifications, the probability that a given neuron spikes in a time period may be described by
where is a synaptic weight, describing the influence of neuron on neuron, expresses the leak, and provides the spiking history of neuron before, according to

Exponential integrate-and-fire

In the exponential integrate-and-fire model, spike generation is exponential, following the equation:
where is the membrane potential, is the membrane potential threshold, is the membrane time constant, is the resting potential, and is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. Once the membrane potential crosses, it diverges to infinity in finite time.

FitzHugh–Nagumo

Sweeping simplifications to Hodgkin–Huxley were introduced by FitzHugh and Nagumo in 1961 and 1962. Seeking to describe "regenerative self-excitation" by a nonlinear positive-feedback membrane voltage and recovery by a linear negative-feedback gate voltage, they developed the model described by
where we again have a membrane-like voltage and input current with a slower general gate voltage and experimentally-determined parameters. Although not clearly derivable from biology, the model allows for a simplified, immediately available dynamic, without being a trivial simplification.

Morris–Lecar

In 1981 Morris and Lecar combined the Hodgkin–Huxley and FitzHugh–Nagumo models into a voltage-gated calcium channel model with a delayed-rectifier potassium channel, represented by
where.

Hindmarsh–Rose

Building upon the FitzHugh–Nagumo model, Hindmarsh and Rose proposed in 1984 a model of neuronal activity described by three coupled first order differential equations:
with, and so that the variable only changes very slowly. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the variable of the model, which include chaotic dynamics. This makes the Hindmarsh–Rose 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.

Cable theory

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 is defined as
where 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 is the total surface area of the tree of total length, and is its total electrotonic length. For an entire neuron in which the cell body conductance is and the membrane conductance per unit area is, we find the total neuron conductance for dendrite trees by adding up all tree and soma conductances, given by
where we can find the general correction factor experimentally by noting.

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 and diameter which connects with fixed resistance to any number of branching cylinders. We define the conductance ratio of the th cylinder as, where and 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, 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, where the conductance ratio is. Then our total neuron conductance is given by
An example of a compartmental model of a neuron, with an algorithm to reduce the number of compartments and yet retain the salient electrical characteristics, can be found in.

Natural input stimulus neuron models

The models in this category were derived following experiments involving natural stimulation such as light, sound, touch, or odor. In these experiments, the spike pattern resulting from each stimulus presentation varies from trial to trial, but the averaged response from several trials often converges to a clear pattern. Consequently, the models in this category generate a probabilistic relationship between the input stimulus to spike occurrences.

The non-homogeneous Poisson process model (Siebert)

Siebert modeled the neuron spike firing pattern using a non-homogeneous Poisson process model, following experiments involving the auditory system. According to Siebert, the probability of a spiking event at the time interval is proportional to a non negative function, where is the raw stimulus.:
Siebert considered several functions as, including for low stimulus intensities.
The main advantage of Siebert's model is its simplicity. The shortcomings of the model is its inability to reflect properly the following phenomena:
These shortcoming are addressed by the two state Markov Model.

The two state Markov model (Nossenson & Messer)

The spiking neuron model by Nossenson & Messer produces the probability of the neuron to fire a spike as a function of either an external or pharmacological stimulus. The model consists of a cascade of a receptor layer model and a spiking neuron model, as shown in Fig 4. The connection between the external stimulus to the spiking probability is made in two steps: First, a receptor cell model translates the raw external stimulus to neurotransmitter concentration, then, a spiking neuron model connects between neurotransmitter concentration to the firing rate. Thus, the spiking neuron model by itself depends on neurotransmitter concentration at the input stage.
An important feature of this model is the prediction for neurons firing rate pattern which captures, using a low number of free parameters, the characteristic edge emphasized response of neurons to a stimulus pulse, as shown in Fig. 5. The firing rate is identified both as a normalized probability for neural spike firing, and as a quantity proportional to the current of neurotransmitters released by the cell. The expression for the firing rate takes the following form:
where,
P0 could be generally calculated recursively using Euler method, but in the case of a pulse of stimulus it yields a simple closed form expression.
, with being short temporal average of stimulus power.
Other predictions by this model include:
1) The averaged Evoked Response Potential due to population of many neurons in unfiltered measurements resembles the firing rate.
2) The voltage variance of activity due to multiple neuron activity resembles the firing rate.
3) The inter-spike-interval probability distribution takes the form a gamma-distribution like function.
Property of the Model by Nossenson & MesserReferencesDescription of experimental evidence
The shape of the firing rate in response to an auditory stimulus pulseThe Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a visual stimulus pulseThe Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to an olfactory stimulus pulseThe Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a somato-sensory stimulusThe Firing Rate has the same shape of Fig 5.
The change in firing rate in response to neurotransmitter application Firing Rate change in response to neurotransmitter application
Square dependence between an auditory stimulus pressure and the firing rateSquare Dependence between Auditory Stimulus pressure and the Firing Rate.
Square dependence between visual stimulus electric field and the firing rateSquare dependence between visual stimulus electric field - Linear Dependence between Visual Stimulus Power and the Firing Rate.
The shape of the Inter-Spike-Interval Statistics ISI shape resembles the gamma-function-like
The ERP resembles the firing rate in unfiltered measurementsThe shape of the averaged evoked response potential in response to stimulus resembles the firing rate.
MUA power resembles the firing rateThe shape of the empirical variance of extra-cellular measurements in response to stimulus pulse resembles the firing rate.

Non-Markovian models

The following is a list of published non-Markovian neuron models:
The theta model, or Ermentrout–Kopell canonical model, is a model originally developed to model neurons in the animal Aplysia, and is particularly well suited to describe neuron parabolic bursting.

Pharmacological input stimulus neuron models

The models in this category produce predictions for experiments involving pharmacological stimulation.

Synaptic transmission (Koch & Segev)

According to the model by Koch and Segev, the response of a neuron to individual neurotransmitters can be modeled as an extension of the classical Hodgkin–Huxley model with both standard and nonstandard kinetic currents. Four neurotransmitters primarily have influence in the CNS. AMPA/kainate receptors are fast excitatory mediators while NMDA receptors mediate considerably slower currents. Fast inhibitory currents go through GABAA receptors, while GABAB receptors mediate by secondary G-protein-activated potassium channels. This range of mediation produces the following current dynamics:
where is the maximal conductance and is the equilibrium potential of the given ion or transmitter, while 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. For GABAB, is the concentration of the G-protein, and describes the dissociation of G in binding to the potassium gates.
The dynamics of this more complicated model have been well-studied experimentally and produce important results in terms of very quick synaptic potentiation and depression, that is, fast, short-term learning.

The two state Markov model (Nossenson & Messer)

The model by Nossenson and Messer translates neurotransmitter concentration at the input stage to the probability of releasing neurotransmitter at the output stage. For a more detailed description of this model, see the [|Two state Markov model section] above.

HTM neuron model

The HTM neuron model was developed by Jeff Hawkins and researchers at Numenta and is based on a theory called Hierarchical Temporal Memory, originally described in the book On Intelligence. It is based on neuroscience and the physiology and interaction of pyramidal neurons in the neocortex of the human brain.
, the biological neuron, and the HTM neuron.

Applications

The question of neural modeling is at the heart of the following projects:

Electrical retinal prosthesis

Further reading on this subject:

Neurotransmitter-based retinal prosthesis">Visual prosthesis">retinal prosthesis

Further reading on this subject:

[Cochlear implant]

[Brain–computer interface]

Artificial limb control and sensation

Further reading on this subject see:

Relation between artificial and biological neuron models

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 for artificial neurons, which in a neural network often looks like
where is the output of the th neuron, is the th input neuron signal, is the synaptic weight between the neurons and, and is the activation function. While this model has seen success in machine-learning applications, it is a poor model for real neurons, because it lacks the time-dependence that real neuron spikes exhibit. Biological models of the "integrate-and-fire" type take essentially this form; in medical and pharmacological literature they have largely been superseded by kinetic models such as the Hodgkin–Huxley model.
In the case of modelling a biological neuron, physical analogues are used in place of abstractions such as "weight" and "transfer function". A neuron is filled and surrounded with water containing ions, which carry electric charge. The neuron is bound by an insulating cell membrane and can maintain a concentration of charged ions on either side that determines a capacitance. The firing of a neuron involves the movement of ions into the cell that occurs when neurotransmitters cause ion channels on the cell membrane to open. We describe this by a physical time-dependent current. With this comes a change in voltage, or the electrical potential energy difference between the cell and its surroundings, which is observed to sometimes result in a voltage spike called an action potential which travels the length of the cell and triggers the release of further neurotransmitters. The voltage, then, is the quantity of interest and is given by.

Conjectures regarding the role of the neuron in the wider context of the brain principle of operation

The neurotransmitter-based energy detection scheme

The neurotransmitter-based energy detection scheme suggests that the neural tissue chemically executes a Radar-like detection procedure.
As shown in Fig. 6, the key idea of the conjecture is to account neurotransmitter concentration, neurotransmitter generation and neurotransmitter removal rates as the important quantities in executing the detection task, while referring to the measured electrical potentials as a side effect that only in certain conditions coincide with the functional purpose of each step. The detection scheme is similar to a radar like "energy detection" because it includes signal squaring, temporal summation and a threshold switch mechanism,
just like the energy detector, but it also includes a unit that emphasizes stimulus edges and a variable memory length. According to this conjecture, the physiological equivalent of the energy test statistics is neurotransmitter concentration, and the firing rate corresponds to neurotransmitter current. The advantage of this interpretation is that it leads to a unit consistent explanation which allows to bridge between electrophysiological measurements, biochemical measurements and psychophysical results.
The evidence reviewed in suggest the following association between functionality to histological classification:
  1. Stimulus squaring is likely to be performed by receptor cells.
  2. Stimulus edge emphasizing and signal transduction is performed by neurons.
  3. Temporal accumulation of neurotransmitters is performed by glia cells. Short term neurotransmitter accumulation is likely to occur also in some types of neurons.
  4. Logical switching is executed by glia cells, and it results from exceeding a threshold level of neurotransmitter concentration. This threshold crossing is also accompanied by a change in neurotransmitter leak rate.
  5. Physical all-or-non movement switching is due to muscle cells and results from exceeding a certain neurotransmitter concentration threshold on muscle surroundings.
Note that although the electrophysiological signals in Fig.6 are often similar to the functional signal, there are some stages in which the electrical observation is different from the functional purpose of the corresponding step. In particular, Nossenson et al. suggested that glia threshold crossing has a completely different functional operation compared to the radiated electrophysiological signal, and that the latter might only be a side effect of glia break.

General comments regarding the modern perspective of scientific and engineering models