Biological neuron model

Fig 1. A neuronal action potential ("spike"). Note that the amplitude and the exact shape of the action potential can vary according to the exact experimental technique used for acquiring the signal.

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.

Fig 2. "Whole cell" measurement technique, which captures the spiking activity of a single neuron and produces full amplitude action potentials.

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 (e.g. deafness or blindness), motor movement decision making, and continuous limb control. In that sense, biological neural models differ from artificial neuron models that do not presume to predict the outcomes of experiments involving the biological neural tissue (although artificial neuron models are also concerned with execution of perception and estimation tasks). 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:

Electrical input–output membrane voltage models – These models produce a prediction for membrane output voltage as function of electrical stimulation at the input stage (either voltage or current). 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 (also known as "action potential") or a quiescent state. Other models are more detailed and account for sub-cellular processes.
Natural or pharmacological input neuron models – These models were inspired from experiments involving either natural or pharmacological stimulation. The results of these experiment tend to vary from trial to trial, but the averaged response tends to converge to a clear pattern. Accordingly, the output of natural and pharmacological neuron models is the probability of a spike event as function of the input stimulus. Typically, the output probability is normalized (divided by) a time constant, and the resulting normalized probability is called the "firing rate" and has units of Hertz. The models in this category differ in the functional relationship connecting the input stimulus to the output probability. Models that are sub-categorized as Markov models are the simplest and yield the most tractable results.
Fig 3. Extracellular measurement: Captures spikes with lower amplitudes, often from several spiking sources, depending on the size of the electrode and its proximity to the sources. Despite the decreased amplitude levels produced by this technique, it also 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.

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 effects). 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 (see e.g.^[1]).

Integrate-and-fire

One of the earliest models of a neuron was first investigated in 1907 by Louis Lapicque.^[2] A neuron is represented in time by

I(t)=C_{{\mathrm {m}}}{\frac {dV_{{\mathrm {m}}}(t)}{dt}}

which is just the time derivative of the law of capacitance, $Q = CV$ . When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold $V th$ , 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 $t ref$ 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

\,\!f(I)={\frac {I}{C_{{\mathrm {m}}}V_{{\mathrm {th}}}+t_{{\mathrm {ref}}}I}}

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 model

Experimental evidence supporting the model
Property of the H&H model	References
The shape of an individual spike	^[3]^[4]^[5]^[6]
The identity of the ions involved	^[3]^[4]^[5]^[6]
Spike speed across the axon	^[3]

Main article: Hodgkin–Huxley model

The Hodgkin–Huxley model (H&H model) ^[3]^[4]^[5]^[6] is a model of the relationship between ion currents crossing the neuronal cell membrane and the membrane voltage.^[3]^[4]^[5]^[6] The model is based on experiments that allowed to force membrane voltage using an intra-cellular pipette. This model is based on the concept of membrane ion channels and relies on data from the squid giant axon. Hodgkin-Huxley was awarded the 1963 Nobel Prize in Physiology or Medicine for this model.

We note as before our voltage-current relationship, this time generalized to include multiple voltage-dependent currents:

C_{{\mathrm {m}}}{\frac {dV(t)}{dt}}=-\sum _{i}I_{i}(t,V)

Each current is given by Ohm's Law as

I(t,V)=g(t,V)\cdot (V-V_{{\mathrm {eq}}})

where $g (t, V)$ is the conductance, or inverse resistance, which can be expanded in terms of its constant average $ḡ$ and the activation and inactivation fractions $m$ and $h$ , respectively, that determine how many ions can flow through available membrane channels. This expansion is given by

g(t,V)={\bar {g}}\cdot m(t,V)^{p}\cdot h(t,V)^{q}

and our fractions follow the first-order kinetics

{\frac {dm(t,V)}{dt}}={\frac {m_{\infty }(V)-m(t,V)}{\tau _{{\mathrm {m}}}(V)}}=\alpha _{{\mathrm {m}}}(V)\cdot (1-m)-\beta _{{\mathrm {m}}}(V)\cdot m

with similar dynamics for $h$ , where we can use either $τ$ and $m \infty$ or $α$ and $β$ to define our gate fractions.

With such a form, all that remains is to individually investigate each current one wants to include. Typically, these include inward Ca²⁺ and Na⁺ input currents and 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, 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

I(t)-{\frac {V_{{\mathrm {m}}}(t)}{R_{{\mathrm {m}}}}}=C_{{\mathrm {m}}}{\frac {dV_{{\mathrm {m}}}(t)}{dt}}

where $R m$ 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 $I th = V th / R m$ in order to cause the cell to fire, else it will simply leak out any change in potential. The firing frequency thus looks like

f(I)={\begin{cases}0,&I\leq I_{{\mathrm {th}}}\\{[}t_{{\mathrm {ref}}}-R_{{\mathrm {m}}}C_{{\mathrm {m}}}\log(1-{\tfrac {V_{{\mathrm {th}}}}{IR_{{\mathrm {m}}}}}){]}^{{-1}},&I>I_{{\mathrm {th}}}\end{cases}}

which converges for large input currents to the previous leak-free model with refractory period.^[7]

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 ^[8]

I(t)-{\frac {V_{\mathrm {m} }(t)}{R_{\mathrm {m} }}}=C_{\mathrm {m} }{\frac {d^{\alpha }V_{\mathrm {m} }(t)}{d^{\alpha }t}}

Galves-Löcherbach

3D visualization of the Galves-Löcherbach model for biological neural nets. This visualization is set for 4,000 neurons (4 layers with one population of inhibitory neurons and one population of excitatory neurons each) at 180 intervals of time.

Main article: Galves-Löcherbach model

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.^[9] Given the model specifications, the probability that a given neuron $i$ spikes in a time period $t$ may be described by

${\mathop {\mathrm {Prob} }}(X_{t}(i)=1|{\mathcal {F}}_{t-1})=\phi _{i}{\Biggl (}\sum _{j\in I}W_{j\rightarrow i}\sum _{s=L_{t}^{i}}^{t-1}g_{j}(t-s)X_{s}(j),t-L_{t}^{i}{\Biggl )},$

where $W_{j\rightarrow i}$ is a synaptic weight, 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

$L_{t}^{i}={\text{sup}}\{s<t:X_{s}(i)=1\}.$

Exponential integrate-and-fire

Main article: Exponential integrate-and-fire

In the Exponential Integrate-and-Fire, spike generation is exponential, following the equation:

{\frac {dX}{dt}}=\Delta _{T}\exp \left({\frac {X-X_{T}}{\Delta _{T}}}\right)

where $X$ is the membrane potential, $X_{T}$ is the membrane potential threshold, and $\Delta_T$ is the sharpness of action potential initiation, usually around 1 mV for cortical pyramidal neurons. Once the membrane potential crosses $X_{T}$ , it diverges to infinity in finite time.^[10]

FitzHugh–Nagumo

Main article: FitzHugh–Nagumo model

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{array}{rcl}{\dfrac {dV}{dt}}&=&V-V^{3}-w+I_{{\mathrm {ext}}}\\\\\tau {\dfrac {dw}{dt}}&=&V-a-bw\end{array}}

where we again have a membrane-like voltage and input current with a slower general gate voltage $w$ and experimentally-determined parameters $a = -0.7, b = 0.8, τ = 1/0.08$ . Although not clearly derivable from biology, the model allows for a simplified, immediately available dynamic, without being a trivial simplification.^[11]

Morris–Lecar

Main article: Morris–Lecar model

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

{\begin{array}{rcl}C{\dfrac {dV}{dt}}&=&-I_{{\mathrm {ion}}}(V,w)+I\\\\{\dfrac {dw}{dt}}&=&\phi \cdot {\dfrac {w_{{\infty }}-w}{\tau _{{w}}}}\end{array}}

where $I_{{\mathrm {ion}}}(V,w)={\bar {g}}_{{\mathrm {Ca}}}m_{{\infty }}\cdot (V-V_{{\mathrm {Ca}}})+{\bar {g}}_{{\mathrm {K}}}w\cdot (V-V_{{\mathrm {K}}})+{\bar {g}}_{{\mathrm {L}}}\cdot (V-V_{{\mathrm {L}}})$ .^[7]

Hindmarsh–Rose

Main article: Hindmarsh–Rose model

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{array}{rcl}{\dfrac {dx}{dt}}&=&y+3x^{2}-x^{3}-z+I\\\\{\dfrac {dy}{dt}}&=&1-5x^{2}-y\\\\{\dfrac {dz}{dt}}&=&r\cdot (4(x+{\tfrac {8}{5}})-z)\end{array}}

with $r 2 = x 2 + y 2 + z 2$ , and $r \approx 10 -2$ so that the $z$ variable only changes very slowly. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the $x$ 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

Compartmental models

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 ^[13]^[14] modeled the neuron spike firing pattern using a non-homogeneous Poisson process model, following experiments involving the auditory system.^[13]^[14] According to Siebert, the probability of a spiking event at the time interval $[t,t+\Delta _{t}]$ is proportional to a non negative function $g[s(t)]$ , where $s(t)$ is the raw stimulus.:

$P_{spike}(t\in [t',t'+\Delta _{t}])=\Delta _{t}\cdot g[s(t)]$

Siebert considered several functions as $g[s(t)]$ , including $g[s(t)]\propto s^{2}(t)$ 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 edge emphasizing property of the neuron in response to a stimulus pulse.
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 two state Markov Model.^[15]^[16]^[17]

The two state Markov model (Nossenson & Messer)

The spiking neuron model by Nossenson & Messer ^[15]^[16]^[17] produces the probability of the neuron to fire a spike as a function of either an external or pharmacological stimulus.^[15]^[16]^[17] 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.^[15]^[16]^[17]

Fig 4: High level block diagram of the receptor layer and neuron model by Nossenson & Messer.^[15]^[17]

Fig 5. The prediction for the firing rate in response to a pulse stimulus as given by the model by Nossenson & Messer.^[15]^[17]

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:

$R_{fire}(t)={\frac {P_{spike}(t;\Delta _{t})}{\Delta _{t}}}=[y(t)+R_{0}]\cdot P_{0}(t)$

where,

P0 is the probability of the neuron to be "armed" and ready to fire. It is given by the following differential equation:

${\dot {P}}_{0}=-[y(t)+R_{0}+R_{1}]\cdot P_{0}(t)+R_{1}$

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.^[15]^[18]

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:

$y(t)\simeq g_{gain}\cdot \langle s^{2}(t)\rangle$ , with $\langle s^{2}(t)\rangle$ being short temporal average of stimulus power (given in Watt or other energy per time unit).

R0 corresponds to the intrinsic spontaneous firing rate of the neuron.
R1 is the recovery rate of the neuron from refractory state.

Other predictions by this model include:

1) The averaged Evoked Response Potential (ERP) due to population of many neurons in unfiltered measurements resembles the firing rate.^[17]

2) The voltage variance of activity due to multiple neuron activity resembles the firing rate (also known as Multi-Unit-Activity power or MUA).^[16]^[17]

3) The inter-spike-interval probability distribution takes the form a gamma-distribution like function.^[15]^[18]

Experimental evidence supporting the model by Nossenson & Messer^[15]^[16]^[17]
Property of the Model by Nossenson & Messer	References	Description of experimental evidence
The shape of the firing rate in response to an auditory stimulus pulse	^[19]^[20]^[21]^[22]^[23]	The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a visual stimulus pulse	^[24]^[25]^[26]^[27]	The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to an olfactory stimulus pulse	^[28]	The Firing Rate has the same shape of Fig 5.
The shape of the firing rate in response to a somato-sensory stimulus	^[29]	The Firing Rate has the same shape of Fig 5.
The change in firing rate in response to neurotransmitter application (mostly glutamate)	^[30]^[31]	Firing Rate change in response to neurotransmitter application (Glutamate)
Square dependence between an auditory stimulus pressure and the firing rate	^[32]	Square Dependence between Auditory Stimulus pressure and the Firing Rate (- Linear dependence in pressure square (power)).
Square dependence between visual stimulus electric field (volts) and the firing rate	^[25]	Square dependence between visual stimulus electric field (volts) - Linear Dependence between Visual Stimulus Power and the Firing Rate.
The shape of the Inter-Spike-Interval Statistics (ISI)	^[33]	ISI shape resembles the gamma-function-like
The ERP resembles the firing rate in unfiltered measurements	^[34]	The shape of the averaged evoked response potential in response to stimulus resembles the firing rate (Fig. 5).
MUA power resembles the firing rate	^[17]^[35]	The shape of the empirical variance of extra-cellular measurements in response to stimulus pulse resembles the firing rate (Fig. 5).

Non-Markovian models

The following is a list of published non-Markovian neuron models:

Johnson, and Swami^[36]
Berry and Meister^[37]
Kass and Ventura^[38]

Pharmacological input stimulus neuron models

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

Synaptic transmission (Koch & Segev)

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.^[15]^[16]^[17] For a more detailed description of this model, see the Two state Markov model section above.

Applications

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

Electrical retinal prosthesis

Further reading on this subject:^[1]^[39]

Neurotransmitter based retinal prosthesis

Further reading on this subject^[40]^[41]

Artificial limb control and sensation

Further reading on this subject see:^[42]^[43]^[44]

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

Conjecture 1: 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 in artificial neurons, which in a neural network often looks like

$y_{i}=\phi \left(\sum _{j}w_{{ij}}x_{j}\right)$

where $y i$ is the output of the $i$ th neuron, $x j$ is the $j$ th input neuron signal, $w ij$ is the synaptic weight (or strength of connection) between the neurons $i$ and $j$ , and $φ$ is the activation function. While this model has seen success in machine-learning applications, it is a poor model for real (biological) neurons, because it lacks the time-dependence that real neurons exhibit. Some of the earliest biological models took this form until kinetic models such as the Hodgkin–Huxley model became dominant.

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 $C m$ . 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 $I (t)$ . 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 $V m (t)$ .

Conjecture 2: Loops of spiking neurons for decision making

^[45]

Conjecture 3: The neurotransmitter based energy detection scheme

The neurotransmitter based energy detection scheme ^[17]^[18] suggests that the neural tissue chemically executes a Radar-like detection procedure. A list of experimental evidence supporting this conjecture is given in.^[17]^[18] This conjecture attributes active functional roles to non-spiking neurons and glia cells.

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.^[7] 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 attempt to explain these phenomena, but are less developed than older models and have yet to be widely applied. Also improbable possibility of modelling of local chronobiology mechanisms.
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).

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Biological neuron model

Electrical input–output membrane voltage models

Integrate-and-fire

Hodgkin–Huxley model

Leaky integrate-and-fire

Fractional-Order Leaky integrate-and-fire

Galves-Löcherbach

Exponential integrate-and-fire

FitzHugh–Nagumo

Morris–Lecar

Hindmarsh–Rose

Cable theory

Compartmental models

Natural input stimulus neuron models

The non-homogeneous Poisson process model (Siebert)

The two state Markov model (Nossenson & Messer)

Non-Markovian models

Pharmacological input stimulus neuron models

Synaptic transmission (Koch & Segev)

The two state Markov model (Nossenson & Messer)

Applications

Electrical retinal prosthesis

Neurotransmitter based retinal prosthesis

Artificial limb control and sensation

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

Conjecture 1: Relation between artificial and biological neuron models

Conjecture 2: Loops of spiking neurons for decision making

Conjecture 3: The neurotransmitter based energy detection scheme

General comments regarding the modern perspective of scientific and engineering models

See also

References