Biological neuron model

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.).

Integrate-and-fire

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

which is just the time derivative of the law of capacitance, 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

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

The Hodgkin–Huxley model (H&H model) is a model of the relationship between ion currents crossing the neuronal cell membrane and the membrane voltage. 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:

Each current is given by Ohm's Law as

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

and our fractions follow the first-order kinetics

with similar dynamics for h, where we can use either τ and m_∞ 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

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

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

Fractional-Order Leaky integrate-and-fire

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

Galves-Löcherbach

The Galves-Löcherbach model is a specific development of the leaky integrate-and-fire model. It is inherently stochastic. It was developed by mathematicians Antonio Galves and Eva Löcherbach. Given the model specifications, the probability that a given neuron i spikes in a time period t may be described by

P r o b ( X t ( i ) = 1 | F t − 1 ) = ϕ i ( ∑ j ∈ I W j → i ∑ s = L t i t − 1 g j ( t − s ) X s ( j ) , t − L t i ) ,

where W j → 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 = sup { s < t : X s ( i ) = 1 } .

Exponential integrate-and-fire

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

where X is the membrane potential, X T is the membrane potential threshold, and Δ 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.

FitzHugh–Nagumo

Sweeping simplifications to Hodgkin–Huxley were introduced by FitzHugh and Nagumo in 1961 and 1962. Seeking to describe "regenerative self-excitation" by a nonlinear positive-feedback membrane voltage and recovery by a linear negative-feedback gate voltage, they developed the model described by

where we again have a membrane-like voltage and input current with a slower general gate voltage 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.

Morris–Lecar

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

where I i o n ( V , w ) = g ¯ C a m ∞ ⋅ ( V − V C a ) + g ¯ K w ⋅ ( V − V K ) + g ¯ L ⋅ ( V − V L ) .

Hindmarsh–Rose

Building upon the FitzHugh–Nagumo model, Hindmarsh and Rose proposed in 1984 a model of neuronal activity described by three coupled first order differential equations:

with r² = x² + y² + z², and r ≈ 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

Cable theory describes the dendritic arbor as a cylindrical structure undergoing a regular pattern of bifurcation, like branches in a tree. For a single cylinder or an entire tree, the input conductance at the base (where the tree meets the cell body, or any such boundary) is defined as

where L is the electrotonic length of the cylinder which depends on its length, diameter, and resistance. A simple recursive algorithm scales linearly with the number of branches and can be used to calculate the effective conductance of the tree. This is given by

where A_D = πld is the total surface area of the tree of total length l, and L_D is its total electrotonic length. For an entire neuron in which the cell body conductance is G_S and the membrane conductance per unit area is G_md = G_m / A, we find the total neuron conductance G_N for n dendrite trees by adding up all tree and soma conductances, given by

where we can find the general correction factor F_dga experimentally by noting G_D = G_mdA_DF_dga.

Compartmental models

The cable model makes a number of simplifications to give closed analytic results, namely that the dendritic arbor must branch in diminishing pairs in a fixed pattern. A compartmental model allows for any desired tree topology with arbitrary branches and lengths, but makes simplifications in the interactions between branches to compensate. Thus, the two models give complementary results, neither of which is necessarily more accurate.

Each individual piece, or compartment, of a dendrite is modeled by a straight cylinder of arbitrary length l and diameter d which connects with fixed resistance to any number of branching cylinders. We define the conductance ratio of the ith cylinder as B_i = G_i / G_∞, where G ∞ = π d 3 / 2 2 R i R m and R_i 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 B_out,i = B_in,i+1, as

where the last equation deals with parents and daughters at branches, and X i = l i 4 R i d i R m . 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 B_in,stem. Then our total neuron conductance is given by

An example of a compartmental model of a neuron, with an algorithm to reduce the number of compartments (increase the computational speed) and yet retain the salient electrical characteristics, can be found in.

Natural input stimulus neuron models

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

The non-homogeneous Poisson process model (Siebert)

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

P s p i k e ( t ∈ [ t ′ , t ′ + Δ t ] ) = Δ t ⋅ g [ s ( t ) ]

Siebert considered several functions as g [ s ( t ) ] , including g [ s ( t ) ] ∝ 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:

These shortcoming are addressed by the two state Markov Model.

The two state Markov model (Nossenson & Messer)

The spiking neuron model by Nossenson & Messer produces the probability of the neuron to fire a spike as a function of either an external or pharmacological stimulus. The model consists of a cascade of a receptor layer model and a spiking neuron model, as shown in Fig 4. The connection between the external stimulus to the spiking probability is made in two steps: First, a receptor cell model translates the raw external stimulus to neurotransmitter concentration, then, a spiking neuron model connects between neurotransmitter concentration to the firing rate (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 f i r e ( t ) = P s p i k e ( t ; Δ t ) Δ t = [ y ( t ) + R 0 ] ⋅ P 0 ( t )

where,

P ˙ 0 = − [ y ( t ) + R 0 + R 1 ] ⋅ 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 ) ≃ g g a i n ⋅ ⟨ s 2 ( t ) ⟩ , with ⟨ s 2 ( t ) ⟩ being short temporal average of stimulus power (given in Watt or other energy per time unit).

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.

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.

Non-Markovian models

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

Pharmacological input stimulus neuron models

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

Synaptic transmission (Koch & Segev)

According to the model by Koch and Segev, the response of a neuron to individual neurotransmitters can be modeled as an extension of the classical Hodgkin–Huxley model with both standard and nonstandard kinetic currents. Four neurotransmitters primarily have influence in the CNS. AMPA/kainate receptors are fast excitatory mediators while NMDA receptors mediate considerably slower currents. Fast inhibitory currents go through GABA_A receptors, while GABA_B receptors mediate by secondary G-protein-activated potassium channels. This range of mediation produces the following current dynamics:

where ḡ is the maximal conductance (around 1S) and E is the equilibrium potential of the given ion or transmitter (AMDA, NMDA, Cl, or K), while [O] 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 B(V). For GABA_B, [G] is the concentration of the G-protein, and K_d describes the dissociation of G in binding to the potassium gates.

The dynamics of this more complicated model have been well-studied experimentally and produce important results in terms of very quick synaptic potentiation and depression, that is, fast, short-term learning.

The two state Markov model (Nossenson & Messer)

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

Applications

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

Electrical retinal prosthesis

Further reading on this subject:

Neurotransmitter based retinal prosthesis

Further reading on this subject

Artificial limb control and sensation

Further reading on this subject see:

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 = ϕ ( ∑ j w i j x j )

where y_i is the output of the i th neuron, x_j is the jth 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 3: 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:

1. Stimulus squaring is likely to be perform by receptor cells.
2. Stimulus edge emphasizing and signal transduction is performed by neurons.
3. Temporal accumulation (=memory) is performed by glia cells. Short term neurotransmitter accumulation is likely to occur also in some types of neurons.
4. Logical switching is executed by glia cells, and it results from exceeding a threshold level of neurotransmitter concentration. This threshold crossing is also accompanied by a stop in the growth of neurotransmitter leak rate.
5. Physical all-or-non movement switching is due to muscle cells and results from exceeding a certain neurotransmitter concentration threshold on muscle surroundings.

Note that although the electrophysiological signals in Fig.6 are often similar to the functional signal (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