Binding neuron

The binding neuron (SN) is a concept of signal processing in a general type neuron or a mathematical model that implements this concept.

Content

Concept Description

For a neuron of a general type, stimuli are stimuli. As a rule, to excite a neuron to the level when it generates an outgoing pulse, more than one incoming pulse is needed. Let the neuron get $n$ ${\ displaystyle n}$ incoming pulses at successive times $t_{1},t_{2},\dots ,t_{n}$ ${\ displaystyle t_ {1}, t_ {2}, \ dots, t_ {n}}$ . The concept of CH introduces the concept of temporal coherence, $t c$ ${\ displaystyle tc}$ between incoming pulses:

$tc={\frac {1}{t_{n}-t_{1}}}\,.$ ${\ displaystyle tc = {\ frac {1} {t_ {n} -t_ {1}}} \ ,.}$

The high temporal coherence between incoming pulses testifies in favor of the fact that in the external world everything $n$ ${\ displaystyle n}$ pulses can be generated by a single complex event. Accordingly, the CH, stimulated by a set of pulses that is sufficiently coherent in time, generates and sends an outgoing pulse. In the terminology of SN, this is called the binding of elementary events (incoming pulses) into a single event. Binding is realized if the set of stimulating pulses is characterized by sufficient temporal coherence and is not realized if the coherence of the incoming pulses is insufficient.

Inhibition of the concept of SN (primarily refers to slow somatic potassium inhibition) controls the level of temporal coherence between incoming pulses, necessary for linking them into a single event: the higher the degree of inhibition, the greater the temporal coherence is necessary for binding.

The signal processing circuit in the connecting neuron.

t_{1},t_{2},\dots ,t_{n}

{\ displaystyle t_ {1}, t_ {2}, \ dots, t_ {n}}

- moments of receipt of incoming pulses.

An outgoing pulse is considered an abstract representation of a complex event (a set of time-coherent incoming pulses), see Scheme.

Origin

“Although the neuron consumes energy, its main function is to receive signals and send them outside, that is, to process information” —the words of Francis Crick indicate the need to describe the function of an individual neuron in terms of processing abstract signals ^[1] . There are two abstract concepts of the functioning of a neuron: “coincidence detector” and “time integrator” ^[2] ^[3] . In the first of these, it is assumed that the neuron is excited and generates an outgoing pulse if a certain number of incoming pulses arrive simultaneously. The concept of a time integrator assumes that a neuron is excited and sends an outgoing pulse as a result of receiving a certain number of incoming pulses separated in time. Each of these concepts takes into account some features of the functioning of real neurons, since it is known that a particular neuron can act as a coincidence detector or as a time integrator depending on the nature of the incoming stimuli ^[4] . However, it is known that in addition to the excitatory impulses, neurons also receive inhibitory stimulation. The natural development of the above two concepts would be a concept that gives inhibition its own role in signal processing.

At the same time, in neuroscience there is the concept of the . For example, in visual perception, signs such as the shape, color, and spatial arrangement of a visual object are represented in the brain in different groups of neurons. The mechanism that ensures the perception of a combination of these signs as such, which belong to one real object, is called the linking of signs ^[5] . There is an experimentally confirmed idea that for the functioning of the binding mechanism, precise coordination of the moments of nerve impulses corresponding to one or another feature is necessary ^[6] ^[7] ^[8] ^[9] ^[10] ^[11] . This coordination basically assumes that signals of different signs arrive in certain areas of the brain within a certain time window.

The concept of SN at the level of an individual neuron satisfies the requirements formulated earlier at the level of large-scale neural networks to ensure the functioning of the binding mechanism. The concept of CH is formulated on the basis of an analysis of the response of the Hodgkin – Huxley equations to stimuli similar to those obtained in real-life neurons, see the “Mathematical implementations” section below.

Mathematical implementations

The Hodgkin-Huxley Model (XX)

The Hodgkin-Huxley model is a physiologically valid model of a neuron that operates in terms of ion currents through the neuron membrane and describes the mechanism by which a neuron generates an action potential.

In ^[12] , the response of model XX to stimuli was numerically investigated. $U(t)$ ${\ displaystyle U (t)}$ composed of a large number of exciting pulses, the moments of receipt of which are randomly distributed within a certain time window $W$ ${\ displaystyle W}$ :

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

Here $V(t)$ ${\ displaystyle V (t)}$ denotes the magnitude of the exciting postsynaptic potential at time $t$ ${\ displaystyle t}$ , $t_{k}$ ${\ displaystyle t_ {k}}$ - moment of receipt $k$ ${\ displaystyle k}$ th pulse, $N P$ ${\ displaystyle NP}$ - the total number of pulses. Numbers $t_{k}$ ${\ displaystyle t_ {k}}$ - random and evenly distributed on the interval $[0;W]$ ${\ displaystyle [0; W]}$ . The resulting incoming stimulus for neuron X-X was obtained as follows:

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

Where $C_{M}$ ${\ displaystyle C_ {M}}$ - the capacity of a unit of the surface of an excitable membrane. The probability was calculated $f p$ ${\ displaystyle fp}$ excitation of a neuron before spike generation depending on the width of the time window $W$ ${\ displaystyle W}$ . X-X equations with different values of fixed potassium conductivity were considered to create a certain level of braking potential. Dependencies $f p$ ${\ displaystyle fp}$ from $W$ ${\ displaystyle W}$ recalculated by the inverse of the window width, $TC={\frac {1}{W}}$ ${\ displaystyle TC = {\ frac {1} {W}}}$ which is analogous to the temporal coherence of pulses in a stimulus, see Fig. 1. The obtained dependences have the form of a step, and the position of the step is regulated by the amount of inhibition, see Fig. 1. Such a character of dependencies allows us to interpret equations X-X as a mathematical implementation of the concept of CH.

Fig. 1. The dependence of the probability of pulse generation

f p

{\ displaystyle fp}

in a Hodgkin-Huxley type neuron stimulated by an aggregate

N P

{\ displaystyle NP}

incoming pulses, on the degree of their temporal coherence. Curves from left to right correspond to an increase in the potassium conductivity of the membrane, that is, an increasing degree of inhibition.

Integrated lossy neuron (INP)

The lossy integrating neuron model is a widespread abstract neuron model. If for INP to pose a task similar to that described above for model XX, then with properly organized braking, you can get step-like dependencies similar to those shown in Fig. 1. Thus, INP can serve as a mathematical implementation of the concept of SN.

Model "binding neuron"

The model of the connecting neuron implements the concept of HF in the most refined form. In this model, the neuron has an internal memory of fixed duration $\tau$ ${\ displaystyle \ tau}$ . Each incoming pulse is stored neuron unchanged for $\tau$ ${\ displaystyle \ tau}$ units of time, after which disappears. The neuron is also characterized by a threshold value. $N_{th}$ ${\ displaystyle N_ {th}}$ number of pulses stored in memory: if the number of pulses in the memory is equal to or greater than the number $N_{th}$ ${\ displaystyle N_ {th}}$ , then the neuron sends an outgoing pulse and is released from the stored incoming. The presence of inhibition in the CH model is realized as a decrease $\tau$ ${\ displaystyle \ tau}$ .

Internal memory $\tau$ ${\ displaystyle \ tau}$ in the SN model, the temporal length of the excitatory postsynaptic potential that occurs in a real neuron when it receives an incoming pulse (the incoming synaptic current) corresponds to the time. On the other hand, $\tau$ ${\ displaystyle \ tau}$ corresponds to the threshold value of the temporal coherence between incoming pulses, at which the pulse generation is still possible in the CH concept (in Fig. 1 - the position of the step).

In the SN model, when calculating the response of a neuron to a stream of incoming pulses, it is necessary to control the time that remains for every impulse present in the neuron (lifetime) to spend in the neuron. This makes the CH model more difficult for numerical simulation than the INV model, where only total excitation is to be monitored. On the other hand, each pulse spends a finite time in the SN $\tau$ ${\ displaystyle \ tau}$ after which disappears without a trace. This distinguishes the SN model from INP, in which the trace of the incoming pulse can be stored indefinitely and disappears only during the generation of the outgoing pulse. This property of the CH model allows one to accurately describe the statistics of outgoing CH activity during stimulation by a random stream of pulses, see ^[13] ^[14] ^[15] .

Extreme case of CH with infinite memory $\tau \to \infty$ ${\ displaystyle \ tau \ to \ infty}$ corresponds to the time integrator. Extreme case of CH with infinitely short memory $\tau \to 0$ ${\ displaystyle \ tau \ to 0}$ matches the match detector.

Microcircuit implementations

As mentioned above, and other models of neurons and neural networks of them have their own implementation at the level of electronic circuits. Among the used chips, we note the user-programmable valve arrays, FPGA . FPGAs are used to model neurons of any type, but they most naturally implement the CH model, since it can be specified without the use of floating point numbers and does not require solving differential equations. These features of the CH model are used, for example, in ^[16] ^[17] .

Limitations

The concept of CH has a number of limitations that follow from its abstract nature. These constraints include ignoring the morphology of neurons, the identical intensity of the incoming pulses, the replacement of a number of transient processes with different relaxation times observed in real neurons with one lifetime $\tau$ ${\ displaystyle \ tau}$ impulse in the neuron, the absence of a refractory period, consideration of only slow (potassium) inhibition.

The CH model also has the same limitations, although some of them can be eliminated by complicating the model, see, for example, ^[18] , where the CH model with a refractory period and fast inhibition is used.

Literature

↑ F. Crick. The Astonishing Hypothesis. Touchstone. 1995.
↑ M. Abeles. Role of the cortical neuron: integrator or coincidence detector? Israel Journal of Medical Sciences, 18: 83-92, 1982. PMID 6279540
↑ P. König, AK Engel, and W. Singer. Integrator coincidence detector? the role of the cortical neuron revisited. Trends in Neurosciences, 19 (4): 130-137, 1996. https://dx.doi.org/10.1016/S0166-2236(96)80019-1 PMID 8658595
↑ M. Rudolph and A. Destexhe. Tuning neocortical pyramidal neurons between integrators and coincidence detectors. Journal of Computational.Neuroscience, 14 (3): 239-251, 2003. https://dx.doi.org/10.1023/A:1023245625896 PMID 12766426
↑ JP Sougné. Binding problem. In Encyclopedia of Cognitive Science. John Wiley & Sons, Ltd, 2006.
↑ AM Treisman and G. Gelade. A feature-integration theory of attention. Cognitive Psychology, 12: 97-136, 1980. https://dx.doi.org/10.1016/0010-0285(80)90005-5 PMID 7351125
↑ von der C. Malsburg. Binding: The modeler's perspective. Neuron 24 (8): 95-104, 1999. Https://dx.doi.org/10.1016/S0896-6273(00)80825-9 PMID 10677030
↑ R. Eckhorn, R. Bauer, W. Jordan, M. Brosch, W. Kruse, M. Munk, and HJ Reitboeck. Coherent oscillations: a mechanism for feature linking in the visual cortex? Biological Cybernetics, 60: 121-130, 1988. https://dx.doi.org/10.1007/BF00202899 PMID 3228555
↑ ar Damasio. Concepts in the brain. Mind & Language, 4 (1-2): 25-28, 1989. https://dx.doi.org/10.1111/j.1468-0017.1989.tb00236.x
↑ AK Engel, P. König, AK Kreiter, CM Gray, and W. Singer. There is a potential for solution to the binding problem: physiological evidence. In HG Schuster and W. Singer, editors, Nonlinear Dynamics and Neuronal Networks, pages 325. VCH Weinheim, 1991.
↑ MM Merzenich, C. Schreiner, W. Jenkins, and X. Wang. Some aspects of learning disabilities. In P. Tallal, AM Galaburda, RR Llinás, and C. Von Euler, editors, Temporal Information Processing on the Nervous System, page 122. The New York Academy of Sciences, 1993. https://dx.doi.org/10.1111 /j.1749-6632.1993.tb22955.x
↑ AK Vidybida. Neuron as time coherence discriminator. Biological Cybernetics, 74 (6): 537-542, 1996. Https://dx.doi.org/10.1007/BF00209424 PMID 8672560
↑ О. K. Vіdibіda. Vikhіdny pot_v zvyuyuyuchy neuron. Ukrainian Mathematical Journal, 59 (12): 1619-1838, 2007, https://dx.doi.org/10.1007/s11253-008-0028-5
↑ AK Vidybida and KG Kravchuk. Delayed feedback makes neuronal firing statistics non-markovian. Ukrainian Mathematical Journal, 64 (12): 1587-1609, 2012, https://dx.doi.org/10.1007/s11253-013-0753-2
↑ Arunachalam, V., Akhavan-Tabatabaei, R., Lopez, C. Modified Hourglass Model for Neuronal Network. Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 374878, 2013, https://dx.doi.org/10.1155/2013/374878
L JL Rosselló, V. Canals, A. Morro, and A. Oliver. Hardware implementation of stochastic spiking neural networks. International Journal of Neural Systems, 22 (4): 1250014, 2012. https://dx.doi.org/10.1142/S0129065712500141
↑ R. Wang, G. Cohen, KM Stiefel, TJ Hamilton, J. Tapson, and A. van Schaik. An implementation of a polychronous spiking neural network with delay adaptation. Frontiers in Neuroscience, 7 (14), 2013. https://dx.doi.org/10.3389/fnins.2013.00014 PMID 23408739
↑ KG Kravchuk and AK Vidybida. Non-markovian spiking statistics of a neuron with delayed feedback in the presence of a refractoriness. Mathematical Biosciences and Engineering, 11 (1): 81-104, 2014. https://dx.doi.org/10.3934/mbe.2014.11.81

Links

The article was written using the material of the article A. Vidybida. Binding neuron. In: Mehdi Khosrow-Pour (ed). Third Edition, IGI Global, Hershey PA, 2014, pp. 1123–1134. https://dx.doi.org/10.4018/978-1-4666-5888-2.ch107

[1] F. Crick. The Astonishing Hypothesis. Touchstone. 1995.

[2] M. Abeles. Role of the cortical neuron: integrator or coincidence detector? Israel Journal of Medical Sciences, 18: 83-92, 1982. PMID 6279540

[3] P. König, AK Engel, and W. Singer. Integrator coincidence detector? the role of the cortical neuron revisited. Trends in Neurosciences, 19 (4): 130-137, 1996. https://dx.doi.org/10.1016/S0166-2236(96)80019-1 PMID 8658595

[4] M. Rudolph and A. Destexhe. Tuning neocortical pyramidal neurons between integrators and coincidence detectors. Journal of Computational.Neuroscience, 14 (3): 239-251, 2003. https://dx.doi.org/10.1023/A:1023245625896 PMID 12766426

[5] JP Sougné. Binding problem. In Encyclopedia of Cognitive Science. John Wiley & Sons, Ltd, 2006.

[6] AM Treisman and G. Gelade. A feature-integration theory of attention. Cognitive Psychology, 12: 97-136, 1980. https://dx.doi.org/10.1016/0010-0285(80)90005-5 PMID 7351125

[7] von der C. Malsburg. Binding: The modeler's perspective. Neuron 24 (8): 95-104, 1999. Https://dx.doi.org/10.1016/S0896-6273(00)80825-9 PMID 10677030

[8] R. Eckhorn, R. Bauer, W. Jordan, M. Brosch, W. Kruse, M. Munk, and HJ Reitboeck. Coherent oscillations: a mechanism for feature linking in the visual cortex? Biological Cybernetics, 60: 121-130, 1988. https://dx.doi.org/10.1007/BF00202899 PMID 3228555

[9] r Damasio. Concepts in the brain. Mind & Language, 4 (1-2): 25-28, 1989. https://dx.doi.org/10.1111/j.1468-0017.1989.tb00236.x

[10] AK Engel, P. König, AK Kreiter, CM Gray, and W. Singer. There is a potential for solution to the binding problem: physiological evidence. In HG Schuster and W. Singer, editors, Nonlinear Dynamics and Neuronal Networks, pages 325. VCH Weinheim, 1991.

[11] MM Merzenich, C. Schreiner, W. Jenkins, and X. Wang. Some aspects of learning disabilities. In P. Tallal, AM Galaburda, RR Llinás, and C. Von Euler, editors, Temporal Information Processing on the Nervous System, page 122. The New York Academy of Sciences, 1993. https://dx.doi.org/10.1111 /j.1749-6632.1993.tb22955.x

[12] AK Vidybida. Neuron as time coherence discriminator. Biological Cybernetics, 74 (6): 537-542, 1996. Https://dx.doi.org/10.1007/BF00209424 PMID 8672560

[13] О. K. Vіdibіda. Vikhіdny pot_v zvyuyuyuchy neuron. Ukrainian Mathematical Journal, 59 (12): 1619-1838, 2007, https://dx.doi.org/10.1007/s11253-008-0028-5

[14] AK Vidybida and KG Kravchuk. Delayed feedback makes neuronal firing statistics non-markovian. Ukrainian Mathematical Journal, 64 (12): 1587-1609, 2012, https://dx.doi.org/10.1007/s11253-013-0753-2

[15] Arunachalam, V., Akhavan-Tabatabaei, R., Lopez, C. Modified Hourglass Model for Neuronal Network. Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 374878, 2013, https://dx.doi.org/10.1155/2013/374878

[16] L JL Rosselló, V. Canals, A. Morro, and A. Oliver. Hardware implementation of stochastic spiking neural networks. International Journal of Neural Systems, 22 (4): 1250014, 2012. https://dx.doi.org/10.1142/S0129065712500141

[17] R. Wang, G. Cohen, KM Stiefel, TJ Hamilton, J. Tapson, and A. van Schaik. An implementation of a polychronous spiking neural network with delay adaptation. Frontiers in Neuroscience, 7 (14), 2013. https://dx.doi.org/10.3389/fnins.2013.00014 PMID 23408739

[18] KG Kravchuk and AK Vidybida. Non-markovian spiking statistics of a neuron with delayed feedback in the presence of a refractoriness. Mathematical Biosciences and Engineering, 11 (1): 81-104, 2014. https://dx.doi.org/10.3934/mbe.2014.11.81