Binding neuron

Hodgkin–Huxley model — physiologically substantiated neuronal model, which operates in terms of transmembrane ionic currents, and describes mechanism of generation of action potential.

In the paper [14] the response of the H-H model was studied numerically to stimuli ${\displaystyle U(t)}$ composed of many excitatory impulses distributed randomly within a time window ${\displaystyle W}$:

${\displaystyle U(t)=\sum _{k=1}^{NP}V(t-t_{k}),\qquad t_{k}\in [0;W].}$

Here ${\displaystyle V(t)}$ denotes magnitude of excitatory postsynaptic potential at moment ${\displaystyle t}$; ${\displaystyle t_{k}}$ — is the moment of arrival of ${\displaystyle k}$-th impulse; ${\displaystyle NP}$ — is the total number of impulses the stimulus is composed of. The numbers ${\displaystyle t_{k}}$ are random, distributed uniformly within interval ${\displaystyle [0;W]}$. The stimulating current applied in the H-H equations is as follows

${\displaystyle I(t)=-C_{M}\ {\frac {dU(t)}{dt}},}$

where ${\displaystyle C_{M}}$ — is the capacity of unit area of excitable membrane. The probability to generate action potential was calculated as a function of the window width ${\displaystyle W}$. Different constant potassium conductances were added to the H-H equations in order to create certain levels of inhibitory potential. The dependencies obtained, if recalculated as functions of ${\displaystyle TC={\frac {1}{W}}}$, which is analogous to temporal coherence of impulses in the compound stimulus, have step-like form. The location of the step is controlled by the level of inhibition potential, see Fig. 1. Due to this type of dependence, the H-H equations can be treated as mathematical model of the BN concept.

Fig. 1. Firing probability (${\displaystyle fp}$) of Hodgkin–Huxley type neuron, stimulated with the set of ${\displaystyle NP}$ input impulses as a function of temporal coherence of the impulses. The curves from the left to the right correspond to increasing potassium conductance, that is to increasing degree of inhibition.

Leaky integrate and fire neuron is a widely used abstract neuronal model. If to state a similar problem for the LIF neuron with appropriately chosen inhibition mechanism, then it is possible to obtain step-like dependencies similar to the Fig. 1 as well. Therefore, the LIF neuron as well can be considered as mathematical model of the BN concept.

The binding neuron model implements the BN concept in the most refined form. [15] In this model each input impulse is stored in the neuron during fixed time ${\displaystyle \tau }$ and then disappears. This kind of memory serves as surrogate of the excitatory postsynaptic potential. The model has a threshold ${\displaystyle N_{th}}$: if the number of stored in the BN impulses exceeds ${\displaystyle N_{th}}$ then the neuron fires a spike and clears it internal memory. The presence of inhibition results in the decreased ${\displaystyle \tau }$. In the BN model, it is necessary to control the time to live of any stored impulse during calculation of the neuron's response to input stimulation. This makes the BN model more complicated for numerical simulation than the LIF model. On the other hand, any impulse spends finite time ${\displaystyle \tau }$ in the BN model neuron. This is in contrast to the LIF model, where traces of any impulse can be present infinitely long. This property of the BN model allows to get precise description of output activity of BN stimulated with random stream of input impulses, see [16] [17] .[18]

The limiting case of BN with infinite memory, τ→∞, corresponds to the temporal integrator. The limiting case of BN with infinitely short memory, τ→0, corresponds to the coincidence detector.