Biological neuron model

Biological neuron models, also known as spiking neuron models,^[1] are mathematical descriptions of the conduction of electrical signals in neurons. Neurons (or nerve cells) are electrically excitable cells within the nervous system, able to fire electric signals, called action potentials, across a neural network. These mathematical models describe the role of the biophysical and geometrical characteristics of neurons on the conduction of electrical activity.

Central to these models is the description of how the membrane potential (that is, the difference in electric potential between the interior and the exterior of a biological cell) across the cell membrane changes over time. In an experimental setting, stimulating neurons with an electrical current generates an action potential (or spike), that propagates down the neuron's axon. This axon can branch out and connect to a large number of downstream neurons at sites called synapses. At these synapses, the spike can cause the release of neurotransmitters, which in turn can change the voltage potential of downstream neurons. This change can potentially lead to even more spikes in those downstream neurons, thus passing down the signal. As many as 85% of neurons in the neocortex, the outermost layer of the mammalian brain, consist of excitatory pyramidal neurons,^[2]^[3] and each pyramidal neuron receives tens of thousands of inputs from other neurons.^[4] Thus, spiking neurons are a major information processing unit of the nervous system.

One such example of a spiking neuron model may be a highly detailed mathematical model that includes spatial morphology. Another may be a conductance-based neuron model that views neurons as points and describes the membrane voltage dynamics as a function of trans-membrane currents. A mathematically simpler "integrate-and-fire" model significantly simplifies the description of ion channel and membrane potential dynamics (initially studied by Lapique in 1907).^[5]^[6]

It is easier to obtain experimentally;

It is robust and lasts for a longer time;

It can reflect the dominant effect, especially when conducted in an anatomical region with many similar cells.

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, 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.

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, one or more electrodes are placed 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:

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:

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 measurement effects and interactions of many cells (network effects).

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).^[7] Modeling helps to analyze experimental data and address questions. Models are also important in the context of restoring lost brain functionality through neuroprosthetic devices.

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).

$I_{\mathrm {AMPA} }(t,V)={\bar {g}}_{\mathrm {AMPA} }\cdot [O]\cdot (V(t)-E_{\mathrm {AMPA} })$

$I_{\mathrm {NMDA} }(t,V)={\bar {g}}_{\mathrm {NMDA} }\cdot B(V)\cdot [O]\cdot (V(t)-E_{\mathrm {NMDA} })$

$I_{\mathrm {GABA_{A}} }(t,V)={\bar {g}}_{\mathrm {GABA_{A}} }\cdot ([O_{1}]+[O_{2}])\cdot (V(t)-E_{\mathrm {Cl} })$

$I_{\mathrm {GABA_{B}} }(t,V)={\bar {g}}_{\mathrm {GABA_{B}} }\cdot {\tfrac {[G]^{n}}{[G]^{n}+K_{\mathrm {d} }}}\cdot (V(t)-E_{\mathrm {K} })$

$B_{\mathrm {out} ,i}={\frac {B_{\mathrm {in} ,i+1}(d_{i+1}/d_{i})^{3/2}}{\sqrt {R_{\mathrm {m} ,i+1}/R_{\mathrm {m} ,i}}}}$

$B_{\mathrm {in} ,i}={\frac {B_{\mathrm {out} ,i}+\tanh X_{i}}{1+B_{\mathrm {out} ,i}\tanh X_{i}}}$

$B_{\mathrm {out,par} }={\frac {B_{\mathrm {in,dau1} }(d_{\mathrm {dau1} }/d_{\mathrm {par} })^{3/2}}{\sqrt {R_{\mathrm {m,dau1} }/R_{\mathrm {m,par} }}}}+{\frac {B_{\mathrm {in,dau2} }(d_{\mathrm {dau2} }/d_{\mathrm {par} })^{3/2}}{\sqrt {R_{\mathrm {m,dau2} }/R_{\mathrm {m,par} }}}}+\ldots$

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.^[105]^[65] 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 arrives 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 calculated from a cable equation.

Let us consider a cell membrane in the form of 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 time

We introduce a length scale $\lambda ^{2}={r_{m}}/{r_{l}}$ on the left side and time constant $\tau =c_{m}r_{m}$ on the right side. The cable equation 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, 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

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 $G D = G md A D F dga$ .

The linear cable model makes several 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^[65] 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 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 $i$ th cylinder as $B i = G i / G \infty$ , where $G_{\infty }={\tfrac {\pi d^{3/2}}{2{\sqrt {R_{i}R_{m}}}}}$ 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 $X_{i}={\tfrac {l_{i}{\sqrt {4R_{i}}}}{\sqrt {d_{i}R_{m}}}}$ . 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 for static input is given by

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.^[105]^[106] For static inputs, it is sometimes possible to reduce the number of compartments (increase the computational speed) and yet retain the salient electrical characteristics.^[107]

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

The neurotransmitter-based energy detection scheme[edit]

The neurotransmitter-based energy detection scheme^[74]^[81] 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 for 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^[74]^[81] suggest the following association between functionality to histological classification:

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 differs 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.

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 is not compatible with models that rely on modeling the membrane as a resistance that 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.

[17]

Modern views regarding 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.

[108]

(W. Gerstner, W. Kistler, R. Naud, L. Paninski, Cambridge University Press, 2014).^[27] In particular, Chapters 6 - 10, html online version.

Biological neuron model

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

The neurotransmitter-based energy detection scheme[edit]

[17]

[108]

Neuronal Dynamics: from single neurons to networks and models of cognition

Spiking Neuron Models

Binding neuron

Bayesian approaches to brain function

Brain-computer interfaces

Free energy principle

Models of neural computation

Neural coding

Neural oscillation

Quantitative models of the action potential

Spiking Neural Network