Biological neuron model

A biological neuron model (also known as spiking neuron model) is a mathematical description of the properties of nerve cells, or neurons, that is designed to accurately describe and predict biological processes. This is in contrast to the artificial neuron, which aims for computational effectiveness, although these goals sometimes overlap.

Artificial neuron abstraction

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.

Biological abstraction

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 $V m (t)$ .

Integrate-and-fire

One of the earliest models of a neuron was first investigated in 1907 by Louis Lapicque.^[1] A neuron is represented in time by

I(t)=C_\mathrm{m} \frac{d V_\mathrm{m}(t)}{d t}

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.

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{d V_\mathrm{m} (t)}{d t}

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 \le I_\mathrm{th} \\ {[} t_\mathrm{ref}-R_\mathrm{m} C_\mathrm{m} \log(1-\tfrac{V_\mathrm{th}}{I R_\mathrm{m}}) {]}^{-1}, & I > I_\mathrm{th} \end{cases}

which converges for large input currents to the previous leak-free model with refractory period.^[2]

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.^[3]

Hodgkin–Huxley

Main article: Hodgkin–Huxley model

The most successful and widely used models of neurons have been based on the Markov kinetic model developed from Hodgkin and Huxley's 1952 work based on data from the squid giant axon. We note as before our voltage-current relationship, this time generalized to include multiple voltage-dependent currents:

C_\mathrm{m} \frac{d V(t)}{d t} = -\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{d m(t,V)}{d t} = \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.

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{d V}{d t} &=& V-V^3 - w + I_\mathrm{ext} \\ \\ \tau \dfrac{d w}{d t} &=& V-a-b w \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.^[4]

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{d V}{d t} &=& -I_\mathrm{ion}(V,w) + I \\ \\ \dfrac{d w}{d t} &=& \phi \cdot \dfrac{w_{\infty} - w}{\tau_{w}} \end{array}

where $I_\mathrm{ion}(V,w) = \bar{g}_\mathrm{Ca} m_{\infty}\cdot(V-V_\mathrm{Ca}) + \bar{g}_\mathrm{K} w\cdot(V-V_\mathrm{K}) + \bar{g}_\mathrm{L}\cdot(V-V_\mathrm{L})$ .^[2]

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{d x}{d t} &=& y+3x^2-x^3-z+I \\ \\ \dfrac{d y}{d t} &=& 1-5x^2-y \\ \\ \dfrac{d z}{d t} &=& 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.

Expanded neuron models

While the success of integrating and kinetic models is undisputed, much has to be determined experimentally before accurate predictions can be made. The theory of neuron integration and firing (response to inputs) is therefore expanded by accounting for the nonideal conditions of cell structure.

Cable theory

Compartmental models

Synaptic transmission

Other conditions

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.^[2] 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.

References

↑ Abbott, L.F. (1999). "Lapique's introduction of the integrate-and-fire model neuron (1907)" (PDF). Brain Research Bulletin 50 (5/6): 303–304. doi:10.1016/S0361-9230(99)00161-6. PMID 10643408. Retrieved 2007-11-24.
1 2 3 4 Koch, Christof; Segev, Idan (1999). Methods in neuronal modeling : from ions to networks (2nd ed.). Cambridge, Massachusetts: MIT Press. p. 687. ISBN 0-262-11231-0.
↑ Ostojic, S.; Brunel, N.; Hakim, V. (2009). "How Connectivity, Background Activity, and Synaptic Properties Shape the Cross-Correlation between Spike Trains". Journal of Neuroscience 29 (33): 10234–10253. doi:10.1523/JNEUROSCI.1275-09.2009. PMID 19692598.
↑ Fitzhugh, R.; Izhikevich, E. (2006). "FitzHugh-Nagumo model". Scholarpedia 1 (9): 1349. doi:10.4249/scholarpedia.1349.
↑ Forrest MD (April 2015). "Simulation of alcohol action upon a detailed Purkinje neuron model and a simpler surrogate model that runs >400 times faster". BMC Neuroscience 16 (27). doi:10.1186/s12868-015-0162-6.
↑ Hodgkin, A. L.; Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of Physiology 117 (4): 500–544. doi:10.1113/jphysiol.1952.sp004764. PMC 1392413. PMID 12991237.

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