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Biological neuron models, also known as a spiking neuron models, are mathematical descriptions of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecond in duration, called action potentials or spikes (Fig. 2). Since spikes are transmitted along the
axon An axon (from Greek ἄξων ''áxōn'', axis), or nerve fiber (or nerve fibre: see spelling differences), is a long, slender projection of a nerve cell, or neuron, in vertebrates, that typically conducts electrical impulses known as action p ...
and
synapse In the nervous system, a synapse is a structure that permits a neuron (or nerve cell) to pass an electrical or chemical signal to another neuron or to the target effector cell. Synapses are essential to the transmission of nervous impulses from ...
s from the sending neuron to many other neurons, spiking
neuron A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa ...
s are considered to be a major information processing unit of the
nervous system In biology, the nervous system is the highly complex part of an animal that coordinates its actions and sensory information by transmitting signals to and from different parts of its body. The nervous system detects environmental changes ...
. Spiking neuron models can be divided into different categories: the most detailed mathematical models are biophysical neuron models (also called Hodgkin-Huxley models) that describe the membrane voltage as a function of the input current and the activation of ion channels. Mathematically simpler are integrate-and-fire models that describe the membrane voltage as a function of the input current and predict the spike times without a description of the biophysical processes that shape the time course of an action potential. Even more abstract models only predict output spikes (but not membrane voltage) as a function of the stimulation where the stimulation can occur through sensory input or pharmacologically. This article provides a short overview of different spiking neuron models and links, whenever possible to experimental phenomena. It includes deterministic and probabilistic models.


Introduction: 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 define the scope of the spiking neuron models. For example,
cochlea The cochlea is the part of the inner ear involved in hearing. It is a spiral-shaped cavity in the bony labyrinth, in humans making 2.75 turns around its axis, the modiolus. A core component of the cochlea is the Organ of Corti, the sensory o ...
r
hair cells Hair cells are the sensory receptors of both the auditory system and the vestibular system in the ears of all vertebrates, and in the lateral line organ of fishes. Through mechanotransduction, hair cells detect movement in their environment. ...
, retinal receptor cells, and
retinal bipolar cell As a part of the retina, bipolar cells exist between photoreceptors (rod cells and cone cells) and ganglion cells. They act, directly or indirectly, to transmit signals from the photoreceptors to the ganglion cells. Structure Bipolar cells are ...
s do not spike. Furthermore, many cells in the nervous system are not classified as neurons but instead are classified as
glia Glia, also called glial cells (gliocytes) or neuroglia, are non-neuronal cells in the central nervous system (brain and spinal cord) and the peripheral nervous system that do not produce electrical impulses. They maintain homeostasis, form myel ...
. 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 an electrode (or array of several electrodes) is located 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: 1) Is easier to obtain experimentally; 2) Is robust and lasts for a longer time; 3) 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 details. Some models in this category predict only the moment of occurrence of output spike (also known as "action potential"); other models are more detailed and account for sub-cellular processes. The models in this category can be either deterministic or probabilistic. #
Natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
stimulus 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 (chemical) 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 measurements effects and interactions of many cells (
network Network, networking and networked may refer to: Science and technology * Network theory, the study of graphs as a representation of relations between discrete objects * Network science, an academic field that studies complex networks Mathematic ...
effects). To accelerate the convergence to a unified theory, we list several models in each category, and where applicable, also references to supporting experiments. 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 (e.g., mechanistic models) to more pragmatic models (e.g., phenomenological models). Modeling helps to analyze experimental data and address questions such as: How are the spikes of a neuron related to sensory stimulation or motor activity such as arm movements? What is the neural code used by the nervous system? 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 (generalized) 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 (see e.g.).


Hodgkin–Huxley

The Hodgkin–Huxley model (H&H 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 In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
describing the behaviour of ion channels that permeate the cell membrane of the
squid giant axon The squid giant axon is the very large (up to 1.5 mm in diameter; typically around 0.5 mm) axon that controls part of the water jet propulsion system in squid. It was first described by L. W. Williams in 1909, but this discovery was f ...
. Hodgkin and Huxley were awarded the 1963 Nobel Prize in Physiology or Medicine for this work. We note the voltage-current relationship, with multiple voltage-dependent currents charging the cell membrane of capacity :C_\mathrm \frac = -\sum_i I_i (t, V). The above equation is the time
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the law of
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
, where the change of the total charge must be explained as the sum over the currents. Each current is given by :I(t,V) = g(t,V)\cdot(V-V_\mathrm) 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 :g(t,V)=\bar\cdot m(t,V)^p \cdot h(t,V)^q and our fractions follow the first-order kinetics :\frac = \frac = \alpha_\mathrm (V)\cdot(1-m) - \beta_\mathrm (V)\cdot m 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 end result can be at the small end 20 parameters which one must estimate or measure for an accurate model. In a model of a complex systems of neurons,
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equatio ...
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 (considered fixed in the original model).


Perfect Integrate-and-fire

One of the earliest models of a neuron is the perfect integrate-and-fire model (also called non-leaky integrate-and-fire), first investigated in 1907 by
Louis Lapicque Louis Édouard Lapicque (1 August 1866 – 6 December 1952) was a French neuroscientist, socialist activist, antiboulangist, dreyfusard and freemason who was very influential in the early 20th century. One of his main contributions was to prop ...
. A neuron is represented by its membrane voltage which evolves in time during stimulation with an input current according :I(t)=C \frac which is just the time
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the law of
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
, . When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold , at which point a
delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
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 start 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 (including dead time). The firing frequency as a function of a constant input current is therefore :\,\! f(I)= \frac . 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 clearly 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 Louis Édouard Lapicque (1 August 1866 – 6 December 1952) was a French neuroscientist, socialist activist, antiboulangist, dreyfusard and freemason who was very influential in the early 20th century. One of his main contributions was to prop ...
, contains, compared to the non-leaky integrate-and-fire model a "leak" term in the membrane potential equation, reflecting the diffusion of ions through the membrane. The model equation looks like : C_\mathrm \frac= I(t)-\frac where is the voltage across the cell membrane and is the membrane resistance. (The non-leaky integrate-and-fire model is retrieved in the limit to infinity, i.e. if the membrane is a perfect insulator). 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 :f(I) = \begin 0, & I \le I_\mathrm \\ \left t_\mathrm-R_\mathrm C_\mathrm \log\left(1-\tfrac\right) \right, & I > I_\mathrm \end which converges for large input currents to the previous leak-free model with refractory period. The model can also be used for inhibitory neurons. The biggest disadvantage of the Leaky integrate-and-fire neuron 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.


Adaptive integrate-and-fire

Neuronal adaptation refers to the fact that even in the presence of a constant current injection into the soma, the intervals between output spikes increase. An adaptive integrate-and-fire neuron model combines the leaky integration of voltage with one or several adaptation variables (see Chapter 6.1. in the textbook Neuronal Dynamics) : \tau_\mathrm \frac = R I(t)- _\mathrm (t) - E_\mathrm R \sum_k w_k : \tau_k \frac = - a_k _\mathrm (t) - E_\mathrm w_k + b_k \tau_k \sum_f \delta (t-t^f) where \tau_m is the membrane time constant, is the adaptation current number, with index ''k'', \tau_k is the time constant of adaptation current , is the resting potential and is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value below the firing threshold. The reset value is one of the important parameters of the model. The simplest model of adaptation has only a single adaptation variable and the sum over k is removed. Integrate-and-fire neurons with one or several adaptation variables can account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting. Moreover, adaptive integrate-and-fire neurons with several adaptation variables are able to predict spike times of cortical neurons under time-dependent current injection into the soma.


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. An advantage of this model is that it can capture adaptation effects with a single variable. The model has the following form :I(t)-\frac = C_\mathrm \frac Once the voltage hits the threshold it is reset. Fractional integration has been used to account for neuronal adaptation in experimental data.


'Exponential integrate-and-fire' and 'adaptive exponential integrate-and-fire'

In the
exponential integrate-and-fire Exponential integrate-and-fire models are compact and computationally efficient nonlinear spiking neuron models with one or two variables. The exponential integrate-and-fire model was first proposed as a one-dimensional model. The most prominent t ...
model, spike generation is exponential, following the equation: : \frac - \frac I(t)= \frac \left E_m-V+\Delta_T \exp \left( \frac \right) \right where V is the membrane potential, V_T is the intrinsic membrane potential threshold, \tau_m is the membrane time constant, E_mis the resting potential, and \Delta_T is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. Once the membrane potential crosses V_T, it diverges to infinity in finite time. In numerical simulation the integration is stopped if the membrane potential hits an arbitrary threshold (much larger than V_T) at which the membrane potential is reset to a value . The voltage reset value is one of the important parameters of the model. Importantly, the right-hand side of the above equation contains a nonlinearity that can be directly extracted from experimental data. In this sense the exponential nonlinearity is strongly supported by experimental evidence. In the adaptive exponential integrate-and-fire neuron the above exponential nonlinearity of the voltage equation is combined with an adaptation variabe w : \tau_m \frac = R I(t) + \left E_m-V+\Delta_T \exp \left( \frac \right) \right- R w : \tau \frac = - a _\mathrm (t) - E_\mathrm w + b \tau \delta (t-t^f) where denotes the adaptation current with time scale \tau. Important model parameters are the voltage reset value , the intrinsic threshold V_T, the time constants \tau and \tau_m as well as the coupling parameters and . The adaptive exponential integrate-and-fire model inherits the experimentally derived voltage nonlinearity of the exponential integrate-and-fire model. But going beyond this model, it can also account for a variety of neuronal firing patterns in response to constant stimulation, including adaptation, bursting and initial bursting. However, since the adaptation is in the form of a current, aberrant hyperpolarization may appear. This problem was solved by expressing it as a conductance.


Stochastic models of membrane voltage and spike timing

The models in this category are generalized integrate-and-fire models that include a certain level of stochasticity. Cortical neurons in experiments are found to respond reliably to time-dependent input, albeit with a small degree of variations between one trial and the next if the same stimulus is repeated. Stochasticity in neurons has two important sources. First, even in a very controlled experiment where input current is injected directly into the soma, ion channels open and close stochastically and this channel noise leads to a small amount of variability in the exact value of the membrane potential and the exact timing of output spikes. Second, for a neuron embedded in a cortical network, it is hard to control the exact input because most inputs come from unobserved neurons somewhere else in the brain. Stochasticity has been introduces into spiking neuron models in two fundamentally different forms: either (i) a noisy input current is added to the differential equation of the neuron model; or (ii) the process of spike generation is noisy. In both cases, the mathematical theory can be developed for continuous time, which is then, if desired for the use in computer simulations, transformed into a discrete-time model. The relation of noise in neuron models to variability of spike trains and neural codes is discussed in
Neural Coding Neural coding (or Neural representation) is a neuroscience field concerned with characterising the hypothetical relationship between the stimulus and the individual or ensemble neuronal responses and the relationship among the electrical activit ...
and in Chapter 7 of the textbook Neuronal Dynamics.


Noisy input model (diffusive noise)

A neuron embedded in a network receives spike input from other neurons. Since the spike arrival times are not controlled by an experimentalist they can be considered as stochastic. Thus a (potentially nonlinear) integrate-and-fire model with nonlinearity f(v) receives two inputs: an input I(t) controlled by the experimentalists and a noisy input current I^(t) that describes the uncontrolled background input. : \tau_m \frac = f(V) + R I(t) + R I^\text(t) Stein's model is the special case of a leaky integrate-and-fire neuron and a stationary white noise current I^(t) = \xi(t) with mean zero and unit variance. In the subthreshold regime, these assumptions yield the equation of the Ornstein–Uhlenbeck process : \tau_m \frac =
_m-V The M-V rocket, also called M-5 or Mu-5, was a Japanese solid-fuel rocket designed to launch scientific satellites. It was a member of the Mu family of rockets. The Institute of Space and Astronautical Science (ISAS) began developing the M-V ...
+ R I(t) + R \xi(t) However, in contrast to the standard Ornstein–Uhlenbeck process, the membrane voltage is reset whenever ''V'' hits the firing threshold . Calculating the interval distribution of the Ornstein–Uhlenbeck model for constant input with threshold leads to a first-passage time problem. Stein's neuron model and variants thereof have been used to fit interspike interval distributions of spike trains from real neurons under constant input current. In the mathematical literature, the above equation of the Ornstein–Uhlenbeck process is written in the form : dV = _m-V + R I(t)\frac + \sigma \, dW where \sigma is the amplitude of the noise input and ''dW'' are increments of a Wiener process. For discrete-time implementations with time step dt the voltage updates are : \Delta V = _m-V + R I(t)\frac + \sigma \sqrty where y is drawn from a Gaussian distribution with zero mean unit variance. The voltage is reset when it hits the firing threshold . The noisy input model can also be used in generalized integrate-and-fire models. For example, the exponential integrate-and-fire model with noisy input reads : \tau_m \frac =E_m-V+\Delta_T \exp \left( \frac \right) + R I(t) + R\xi(t) For constant deterministic input I(t)=I_0 it is possible to calculate the mean firing rate as a function of I_0. This is important because the frequency-current relation (f-I-curve) is often used by experimentalists to characterize a neuron. It is also the transfer function in The leaky integrate-and-fire with noisy input has been widely used in the analysis of networks of spiking neurons. Noisy input is also called 'diffusive noise' because it leads to a diffusion of the subthreshold membrane potential around the noise-free trajectory (Johannesma, The theory of spiking neurons with noisy input is reviewed in Chapter 8.2 of the textbook ''Neuronal Dynamics''.


Noisy output model (escape noise)

In deterministic integrate-and-fire models, a spike is generated if the membrane potential hits the threshold V_. In noisy output models the strict threshold is replaced by a noisy one as follows. At each moment in time t, a spike is generated stochastically with instantaneous stochastic intensity o
'escape rate'
'''' : \rho(t) = f(V(t)-V_) that depends on the momentary difference between the membrane voltage and the threshold V_. A common choice for the 'escape rate' f (that is consistent with biological data) is : f(V-V_) = \frac \exp beta(V-V_ where \tau_0is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and \beta is a sharpness parameter. For \beta\to\infty the threshold becomes sharp and spike firing occurs deterministically at the moment when the membrane potential hits the threshold from below. The sharpness value found in experiments is 1/\beta\approx 4mV which means that neuronal firing becomes non-negligible as soon the membrane potential is a few mV below the formal firing threshold. The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook ''Neuronal Dynamics.'' For models in discrete time, a spike is generated with probability : P_F(t_n) = F (t_n)-V_ that depends on the momentary difference between the membrane voltage at time t_n and the threshold V_. The function F is often taken as a standard sigmoidal F(x) = 0.5 + \tanh(\gamma x)/math> with steepness parameter \gamma, similar to the update dynamics in artificial neural networks. But the functional form of F can also be derived from the stochastic intensity f in continuous time introduced above as F(y_n)\approx 1 - \exp _n\Delta t/math> where y_n = V(t_n)-V_ is the distance to threshold. Integrate-and-fire models with output noise can be used to predict the PSTH of real neurons under arbitrary time-dependent input. For non-adaptive integrate-and-fire neurons, the interval distribution under constant stimulation can be calculated from stationary
renewal theory Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ...
. ''''


Spike response model (SRM)

''main article'': Spike response model The spike response model (SRM) is a general linear model for the subthreshold membrane voltage combined with a nonlinear output noise process for spike generation. The membrane voltage at time ''t'' is : V(t)= \sum_f \eta(t-t^f) + \int_0^\infty \kappa(s) I(t-s)\,ds + V_\mathrm where is the firing time of spike number f of the neuron, is the resting voltage in the absence of input, is the input current at time t-s and \kappa(s) is a linear filter (also called kernel) that describes the contribution of an input current pulse at time t-s to the voltage at time t. The contributions to the voltage caused by a spike at time t^f are described by the refractory kernel \eta(t-t^f). In particular, \eta(t-t^f) describes the reset after the spike and the time course of the spike-afterpotential following a spike. It therefore expresses the consequences of refractoriness and adaptation. The voltage V(t) can be interpreted as the result of an integration of the differential equation of a leaky integrate-and-fire model coupled to an arbitrary number of spike-triggered adaptation variables. Spike firing is stochastic and happens with a time-dependent stochastic intensity (instantaneous rate) : f(V-\vartheta(t)) = \frac \exp beta(V-\vartheta(t) with parameters \tau_0 and \beta and a dynamic threshold \vartheta(t) given by : \vartheta(t)= \vartheta_0 + \sum_f \theta_1(t-t^f) Here \vartheta_0 is the firing threshold of an inactive neuron and \theta_1(t-t^f) describes the increase of the threshold after a spike at time t^f. In case of a fixed threshold, one sets \theta_1(t-t^f)=0. For \beta \to \infty the threshold process is deterministic.'''' The time course of the filters \eta,\kappa,\theta_1 that characterize the spike response model can be directly extracted from experimental data. With optimized parameters the SRM describes the time course of the subthreshold membrane voltage for time-dependent input with a precision of 2mV and can predict the timing of most output spikes with a precision of 4ms. The SRM is closely related to linear-nonlinear-Poisson cascade models (also called Generalized Linear Model). The estimation of parameters of probabilistic neuron models such as the SRM using methods developed for Generalized Linear Models is discussed in Chapter 10 of the textbook ''Neuronal Dynamics''. The name spike response model arises because in a network, the input current for neuron i is generated by the spikes of other neurons so that in the case of a network the voltage equation becomes : V_i(t)= \sum_f \eta_i(t-t_i^f) + \sum_^N w_ \sum_\varepsilon_(t-t_j^) + V_\mathrm where t_j^ are the firing times of neuron j (i.e., its spike train), and \eta_i(t-t^f_i) describes the time course of the spike and the spike after-potential for neuron i, w_ and \varepsilon_(t-t_j^) describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike t_j^ of the presynaptic neuron j. The time course \varepsilon_(s) of the PSP results from the convolution of the postsynaptic current I(t) caused by the arrival of a presynaptic spike from neuron j with the membrane filter \kappa(s).


SRM0

The SRM0 is a stochastic neuron model related to time-dependent nonlinear
renewal theory Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ...
and a simplification of the Spike Renose Model (SRM). The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel \eta(s) there is no summation sign over past spikes: only the ''most recent spike'' (denoted as the time \hat) matters. Another difference is that the threshold is constant. The model SRM0 can be formulated in discrete or continuous time. For example, in continuous time, the single-neuron equation is : V(t)= \eta(t-\hat) + \int_0^\infty \kappa(s) I(t-s) \, ds + V_\mathrm and the network equations of the SRM0 are : V_i(t\mid\hat_i) = \eta_i(t-\hat_i) + \sum_j w_ \sum_f \varepsilon_(t-\hat_i,t-t^f) + V_\mathrm where \hat_i is the ''last firing time neuron'' i. Note that the time course of the postsynaptic potential \varepsilon_ is also allowed to depend on the time since the last spike of neuron i so as to describe a change in membrane conductance during refractoriness. The instantaneous firing rate (stochastic intensity) is : f(V-\vartheta) = \frac \exp beta(V-V_) where V_ is a fixed firing threshold. Thus spike firing of neuron i depends only on its input and the time since neuron i has fired its last spike. With the SRM0, the interspike-interval distribution for constant input can be mathematically linked to the shape of the refractory kernel \eta . Moreover the stationary frequency-current relation can be calculated from the escape rate in combination with the refractory kernel \eta. With an appropriate choice of the kernels, the SRM0 approximates the dynamics of the Hodgkin-Huxley model to a high degree of accuracy. Moreover, the PSTH response to arbitrary time-dependent input can be predicted.


Galves–Löcherbach model

The Galves–Löcherbach model is a
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
neuron model closely related to the spike response model SRM0 and to the leaky integrate-and-fire model. It is inherently
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
and, just like the SRM0 linked to time-dependent nonlinear
renewal theory Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ...
. Given the model specifications, the probability that a given neuron i spikes in a time period t may be described by : \mathop(X_(i) = 1\mid \mathcal_) = \varphi_i \Biggl( \sum_ W_ \sum_^ g_j(t-s) X_s(j),~~~ t-L_t^i \Biggl), where W_ is a
synaptic weight In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in ...
, describing the influence of neuron j on neuron i, g_j expresses the leak, and L_t^i provides the spiking history of neuron i before t, according to : L_t^i =\sup\. Importantly, the spike probability of neuron i depends only on its spike input (filtered with a kernel g_ and weighted with a factor W_) and the timing of its most recent output spike (summarized by t-L_t^i).


Didactic toy models of membrane voltage

The models in this category are highly simplified toy models that qualitatively describe the membrane voltage as a function of input. They are mainly used for didactic reasons in teaching but are not considered valid neuron models for large-scale simulations or data fitting.


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 :\begin \dfrac &=& V-V^3/3 - w + I_\mathrm \\ \tau \dfrac &=& V-a-b w \end 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. The experimental support is weak, but the model is useful as a didactic tool to introduce dynamics of spike generation through phase plane analysis. See Chapter 7 in the textbook ''Methods of Neuronal Modeling''


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 :\begin & C\frac &=& -I_\mathrm(V,w) + I \\ pt& \frac &=& \varphi \cdot \frac \end where I_\mathrm(V,w) = \bar_\mathrm m_\infty \cdot(V-V_\mathrm) + \bar_\mathrm w\cdot(V-V_\mathrm) + \bar_\mathrm\cdot(V-V_\mathrm). The experimental support of the model is weak, but the model is useful as a didactic tool to introduce dynamics of spike generation through phase plane analysis. See Chapter 7 in the textbook ''Methods of Neuronal Modeling''. A two-dimensional neuron model very similar to the Morris-Lecar model can be derived step-by-step starting from the Hodgkin-Huxley model. See Chapter 4.2 in the textbook Neuronal Dynamics.


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: :\begin & \frac &=& y+3x^2-x^3-z+I \\ pt& \frac &=& 1-5x^2-y \\ pt& \frac &=& r\cdot (4(x + \tfrac)-z) \end 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 firing patterns of the action potential, in particular bursting, observed in experiments. Nevertheless, it remains a toy model and has not been fitted to experimental data. It is widely used as a reference model for bursting dynamics.


Theta model and quadratic integrate-and-fire.

The theta model, or Ermentrout–Kopell
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
Type I model, is mathematically equivalent to the quadratic integrate-and-fire model which in turn is an approximation to the exponential integrate-and-fire model and the Hodgkin-Huxley model. It is called a canonical model because it is one of the generic models for constant input close to the bifurcation point, which means close to the transition from silent to repetitive firing. The standard formulation of the theta model is : \frac = (I-I_0) + \cos(\theta)+ - \cos(\theta) The equation for the quadratic integrate-and-fire model is (see Chapter 5.3 in the textbook Neuronal Dynamics )) : \tau_\mathrm \frac = (I-I_0) R + _\mathrm (t) - E_\mathrm V_\mathrm (t) - V_\mathrm ] The equivalence of theta model and quadratic integrate-and-fire is for example reviewed in Chapter 4.1.2.2 of spiking neuron models. For input I(t) that changes over time or is far away from the bifurcation point, it is preferable to work with the exponential integrate-and-fire model (if one wants the stay in the class of one-dimensional neuron models), because real neurons exhibit the nonlinearity of the exponential integrate-and-fire model.


Sensory input-stimulus encoding 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. Importantly, the recorded neurons are often located several processing steps after the sensory neurons, so that these models summarize the effects of the sequence of processing steps in a compact form


The non-homogeneous Poisson process model (Siebert)

Siebert modeled the neuron spike firing pattern using a non-
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
model, following experiments involving the auditory system. According to Siebert, the probability of a spiking event at the time interval , t+\Delta_t/math> is proportional to a non negative function g (t)/math>, where s(t) is the raw stimulus.: : P_\text(t\in ',t'+\Delta_t=\Delta_t \cdot g (t)/math> Siebert considered several functions as g (t)/math>, including g (t)\propto s^2(t) 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: * The transient enhancement of the neuronal firing activity in response to a step stimulus. * The saturation of the firing rate. * The values of inter-spike-interval-histogram at short intervals values (close to zero). These shortcoming are addressed by the age-dependent point process model and the two-state Markov Model.


Refractoriness and age-dependent point process model

Berry and Meister studied neuronal refractoriness using a stochastic model that predicts spikes as a product of two terms, a function f(s(t)) that depends on the time-dependent stimulus s(t) and one a recovery function w(t-\hat) that depends on the time since the last spike : \rho(t) = f(s(t))w(t-\hat) The model is also called an ''inhomogeneous Markov interval (IMI) process''. Similar models have been used for many years in auditory neuroscience. Since the model keeps memory of the last spike time it is non-Poisson and falls in the class of time-dependent renewal models. It is closely related to the model SRM0 with exponential escape rate. Importantly, it is possible to fit parameters of the age-dependent point process model so as to describe not just the PSTH response, but also the interspike-interval statistics.


Linear-nonlinear Poisson cascade model and GLM

The linear-nonlinear-Poisson cascade model is a cascade of a linear filtering process followed by a nonlinear spike generation step. In the case that output spikes feed back, via a linear filtering process, we arrive at a model that is known in the neurosciences as Generalized Linear Model (GLM). The GLM is mathematically equivalent to the spike response model SRM) with escape noise; but whereas in the SRM the internal variables are interpreted as the membrane potential and the firing threshold, in the GLM the internal variables are abstract quantities that summarizes the net effect of input (and recent output spikes) before spikes are generated in the final step.


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 (spiking probability). 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: : R_\text(t)=\frac= (t)+R_0\cdot P_0(t) where, * P0 is the probability of the neuron to be "armed" and ready to fire. It is given by the following differential equation: : \dot_0=- (t)+R_0+R_1\cdot P_0(t) +R_1 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. * ''y''(''t'') is the input of the model and is interpreted as the neurotransmitter concentration on the cell surrounding (in most cases glutamate). For an external stimulus it can be estimated through the receptor layer model: : y(t) \simeq g_\text \cdot \langle s^2(t)\rangle, with \langle s^2(t)\rangle being short temporal average of stimulus power (given in Watt or other energy per time unit). * ''R''0 corresponds to the intrinsic spontaneous firing rate of the neuron. * ''R''1 is the recovery rate of the neuron from the refractory state. Other predictions by this model include: 1) The averaged evoked response potential (ERP) due to the 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 (also known as Multi-Unit-Activity power or MUA). 3) The inter-spike-interval probability distribution takes the form a gamma-distribution like function.


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 receptor The ''N''-methyl-D-aspartate receptor (also known as the NMDA receptor or NMDAR), is a glutamate receptor and ion channel found in neurons. The NMDA receptor is one of three types of ionotropic glutamate receptors, the other two being AMPA and ...
s 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: *I_\mathrm(t,V) = \bar_\mathrm \cdot \cdot (V(t)-E_\mathrm) *I_\mathrm(t,V) = \bar_\mathrm \cdot B(V) \cdot \cdot (V(t)-E_\mathrm) *I_\mathrm(t,V) = \bar_\mathrm \cdot ( _1 _2 \cdot (V(t)-E_\mathrm) *I_\mathrm(t,V) = \bar_\mathrm \cdot \tfrac \cdot (V(t)-E_\mathrm) where is the maximal conductance (around 1 S) and is the equilibrium potential of the given ion or transmitter (AMDA, NMDA, Cl, or K), 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 stochastic 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 Jeffrey Hawkins is a co-founder of the companies Palm Computing, where he co-created the PalmPilot, and Handspring, where he was one of the creators of the Treo.Jeff Hawkins, ''On Intelligence'', p.28 He subsequently turned to work on neurosc ...
and researchers at Numenta and is based on a theory called
Hierarchical Temporal Memory Hierarchical temporal memory (HTM) is a biologically constrained machine intelligence technology developed by Numenta. Originally described in the 2004 book ''On Intelligence'' by Jeff Hawkins with Sandra Blakeslee, HTM is primarily used today for ...
, originally described in the book ''
On Intelligence ''On Intelligence: How a New Understanding of the Brain will Lead to the Creation of Truly Intelligent Machines'' is a 2004 book by Jeff Hawkins and Sandra Blakeslee. The book explains Hawkins' memory-prediction framework theory of the brain a ...
''. It is based on
neuroscience Neuroscience is the science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...
and the physiology and interaction of pyramidal neurons in the
neocortex The neocortex, also called the neopallium, isocortex, or the six-layered cortex, is a set of layers of the mammalian cerebral cortex involved in higher-order brain functions such as sensory perception, cognition, generation of motor commands, sp ...
of the human brain.


Applications

Spiking Neuron Models are used in a variety of applications that need encoding into or decoding from neuronal spike trains in the context of neuroprosthesis and brain-computer interfaces such as
retinal prosthesis A visual prosthesis, often referred to as a bionic eye, is an experimental visual device intended to restore functional vision in those with partial or total blindness. Many devices have been developed, usually modeled on the cochlear implant or ...
: or artificial limb control and sensation. Applications are not part of this article; for more information on this topic please refer to the main article.


Relation between artificial and biological neuron models

The most basic model of a neuron consists of an input with some
synaptic weight In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in ...
vector and an
activation function In artificial neural networks, the activation function of a node defines the output of that node given an input or set of inputs. A standard integrated circuit can be seen as a digital network of activation functions that can be "ON" (1) or " ...
or
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
inside the neuron determining output. This is the basic structure used for artificial neurons, which in a
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
often looks like : y_i = \varphi\left( \sum_j w_ x_j \right) where is the output of the th neuron, is the th input neuron signal, is the synaptic weight (or strength of connection) between the neurons and , and is the
activation function In artificial neural networks, the activation function of a node defines the output of that node given an input or set of inputs. A standard integrated circuit can be seen as a digital network of activation functions that can be "ON" (1) or " ...
. While this model has seen success in machine-learning applications, it is a poor model for real (biological) neurons, because it lacks time-dependence in input and output. When an input is switched on at a time t and kept constant thereafter, biological neurons emit a spike train. Importantly this spike train is not regular but exhibits a temporal structure characterized by adaptation, bursting, or initial bursting followed by regular spiking. Generalized integrate-and-fire model such as the Adaptive Exponential Integrate-and-Fire model, the spike response model, or the (linear) adaptive integrate-and-fire model are able to capture these neuronal firing patterns. Moreover, neuronal input in the brain is time-dependent. Time-dependent input is transformed by complex linear and nonlinear filters into a spike train in the output. Again, the spike response model or the adaptive integrate-and-fire model enable to predict the spike train in the output for arbitrary time-dependent input, whereas an artificial neuron or a simple leaky integrate-and-fire does not. If we take the Hodkgin-Huxley model as a starting point, generalized integrate-and-fire models can be derived systematically in a step-by-step simplification procedure. This has been shown explicitly for the
exponential integrate-and-fire Exponential integrate-and-fire models are compact and computationally efficient nonlinear spiking neuron models with one or two variables. The exponential integrate-and-fire model was first proposed as a one-dimensional model. The most prominent t ...
model and the spike response 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 Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
. The firing of a neuron involves the movement of ions into the cell that occurs when
neurotransmitters A neurotransmitter is a signaling molecule secreted by a neuron to affect another cell across a synapse. The cell receiving the signal, any main body part or target cell, may be another neuron, but could also be a gland or muscle cell. Neurot ...
cause
ion channels Ion channels are pore-forming membrane proteins that allow ions to pass through the channel pore. Their functions include establishing a resting membrane potential, shaping action potentials and other electrical signals by gating the flow of i ...
on the cell membrane to open. We describe this by a physical time-dependent
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
. With this comes a change in
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
, or the electrical potential energy difference between the cell and its surroundings, which is observed to sometimes result in a
voltage spike In electrical engineering, spikes are fast, short duration electrical transients in voltage (voltage spikes), current (current spikes), or transferred energy (energy spikes) in an electrical circuit. Fast, short duration electrical transients ( ...
called an
action potential An action potential occurs when the membrane potential of a specific cell location rapidly rises and falls. This depolarization then causes adjacent locations to similarly depolarize. Action potentials occur in several types of animal cells ...
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 . If the input current is constant, most neurons emit after some time of adaptation or initial bursting a regular spike train. The frequency of regular firing in response to a constant current is described by the frequency-current relation which corresponds to the transfer function \varphi of artificial neural networks. Similarly, for all spiking neuron models the transfer function \varphi can be calculated numerically (or analytically).


Cable theory and compartmental models

All of the above deterministic models are point-neuron models because they do not consider the spatial structure of a neuron. However, the dendrite contributes to transforming input into output. Point neuron models are valid description in three cases. (i) If input current is directly injected into the soma. (ii) If synaptic input arrives predominantly at or close to the soma (closeness is defined by a length scale \lambda introduced below. (iii) If synapse arrive anywhere on the dendrite, but the dendrite is completely linear. In the last case the cable acts as a linear filter; these linear filter properties can be included in the formulation of generalized integrate-and-fire models such as the spike response model. The filter properties can be calculate from a
cable equation Classical cable theory uses mathematical models to calculate the electric current (and accompanying voltage) along passive neurites, particularly the dendrites that receive synaptic inputs at different sites and times. Estimates are made by mo ...
. Let us consider a cell membrane in the form a cylindrical cable. The position on the cable is denoted by x and the voltage across the cell membrane by V. The cable is characterized by a longitudinal resistance r_l per unit length and a membrane resistance r_m . If everything is linear, the voltage changes as a function of timeWe introduce a length scale \lambda^2 = / on the left side and time constant \tau = c_m r_m on the right side. The
cable equation Classical cable theory uses mathematical models to calculate the electric current (and accompanying voltage) along passive neurites, particularly the dendrites that receive synaptic inputs at different sites and times. Estimates are made by mo ...
can now be written in its perhaps best known form: The above cable equation is valid for a single cylindrical cable. Linear cable theory describes the dendritic arbor of a neuron as a cylindrical structure undergoing a regular pattern of
bifurcation Bifurcation or bifurcated may refer to: Science and technology * Bifurcation theory, the study of sudden changes in dynamical systems ** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow * River bifurcation, the ...
, like branches in a tree. For a single cylinder or an entire tree, the static input conductance at the base (where the tree meets the cell body, or any such boundary) is defined as :G_ = \frac, 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 :\,\! G_D = G_m A_D \tanh(L_D) / L_D 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 :G_N = G_S + \sum_^n A_ F_, where we can find the general correction factor experimentally by noting . The linear 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 and that dendrites are linear. A compartmental model allows for any desired tree topology with arbitrary branches and lengths, as well as arbitrary nonlinearities. It is essentially a discretized computational implementation of nonlinear dendrites. 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 G_\infty=\tfrac 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 *B_ = \frac *B_ = \frac *B_\mathrm = \frac + \frac + \ldots where the last equation deals with ''parents'' and ''daughters'' at branches, and X_i = \tfrac. 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 . Then our total neuron conductance for static input is given by :G_N = \frac + \sum_j B_ G_. Importantly, static input is a very special case. In biology inputs are time dependent. Moreover, dendrites are not always linear. Compartmental models enable to include nonlinearities via ion channels positioned at arbitrary locations along the dendrites. For static inputs, it is sometimes possible to reduce the number of compartments (increase the computational speed) and yet retain the salient electrical characteristics.


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 (variable memory). 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: # Stimulus squaring is likely to be performed by receptor cells. # Stimulus edge emphasizing and signal transduction is performed by neurons. # Temporal accumulation of neurotransmitters is performed by glial cells. Short term neurotransmitter accumulation is likely to occur also in some types of neurons. # Logical switching is executed by glial 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. # 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 (signal power / neurotransmitter concentration / muscle force), 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

* The models above are still idealizations. Corrections must be made for the increased membrane surface area given by numerous dendritic spines, temperatures significantly hotter than room-temperature experimental data, and nonuniformity in the cell's internal structure. Certain observed effects do not fit into some of these models. For instance, the temperature cycling (with minimal net temperature increase) of the cell membrane during action potential propagation not compatible with models which rely on modeling the membrane as a resistance which must dissipate energy when current flows through it. The transient thickening of the cell membrane during action potential propagation is also not predicted by these models, nor is the changing capacitance and voltage spike that results from this thickening incorporated into these models. The action of some anesthetics such as inert gases is problematic for these models as well. New models, such as the
soliton model The soliton hypothesis in neuroscience is a model that claims to explain how action potentials are initiated and conducted along axons based on a thermodynamic theory of nerve pulse propagation. It proposes that the signals travel along the ce ...
attempt to explain these phenomena, but are less developed than older models and have yet to be widely applied. * Modern views regarding of the role of the scientific model suggest that "All models are wrong but some are useful" (Box and Draper, 1987, Gribbin, 2009; Paninski et al., 2009). *Recent conjecture suggests that each neuron might function as a collection of independent threshold units. It is suggested that a neuron could be anisotropically activated following the origin of its arriving signals to the membrane, via its dendritic trees. The spike waveform was also proposed to be dependent on the origin of the stimulus.


External links


Neuronal Dynamics: from single neurons to networks and models of cognition
(W. Gerstner, W. Kistler, R. Naud, L. Paninski, ''Cambridge University Press, 2014)''. In particular



ref name="Gerstner_2002" /> (W. Gerstner and W. Kistler, Cambridge University Press, 2002)


See also

*
Binding neuron A binding neuron (BN) is an abstract concept of processing of input impulses in a generic neuron based on their temporal coherence and the level of neuronal inhibition. Mathematically, the concept may be implemented by most neuronal models includin ...
*
Bayesian approaches to brain function Bayesian approaches to brain function investigate the capacity of the nervous system to operate in situations of uncertainty in a fashion that is close to the optimal prescribed by Bayesian statistics. This term is used in behavioural sciences and ...
* Brain-computer interfaces *
Free energy principle The free energy principle is a mathematical principle in biophysics and cognitive science that provides a formal account of the representational capacities of physical systems: that is, why things that exist look as if they track properties of the ...
*
Models of neural computation Models of neural computation are attempts to elucidate, in an abstract and mathematical fashion, the core principles that underlie information processing in biological nervous systems, or functional components thereof. This article aims to provide ...
*
Neural coding Neural coding (or Neural representation) is a neuroscience field concerned with characterising the hypothetical relationship between the stimulus and the individual or ensemble neuronal responses and the relationship among the electrical activit ...
*
Neural oscillation Neural oscillations, or brainwaves, are rhythmic or repetitive patterns of neural activity in the central nervous system. Neural tissue can generate oscillatory activity in many ways, driven either by mechanisms within individual neurons or by ...
* Quantitative models of the action potential *
Spiking Neural Network Spiking neural networks (SNNs) are artificial neural networks that more closely mimic natural neural networks. In addition to neuronal and synaptic state, SNNs incorporate the concept of time into their operating model. The idea is that neuron ...


References

{{DEFAULTSORT:Biological Neuron Model Biophysics Computational neuroscience Neuroscience