Realization of spiking by an excitable chemical system

Abstract

It is theoretically demonstrated that an excitable chemical system can function as a spiking neuron and a chemical spiking neuron network can be constructed. The Oregonator is used as a model for a chemical spiking neuron. The chemical spiking neuron modeled by the Oregonator is similar in behavior to the spiking neuron model of Maass even though the Oregonator has a different excitation mechanism from the spiking neuron model. In the spiking neuron network, information is encoded and processed by using the timing of the spikes in spiking neurons. It is shown that chemical spiking neuron networks can be constructed by the unidirectional selective coupling of the spiking chemical neurons. The chemical spiking neuron network can process information encoded temporally like the spiking neuron network with a restriction in the range of the weight of couplings between the chemical neurons.

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1 Introduction

It has been found that nonlinear chemical systems can be used as computational units of networks that perform logic functions, pattern recognition etc.^1–10 The computational abilities of the chemical systems are based on their nonlinear dynamical features such as bistability,^1–5 excitability^8,9etc.¹⁰ When perturbations on parameters or variables as inputs are applied, the nonlinear chemical systems are able to change their states or dynamical behaviors distinctively. When a network is composed of nonlinear chemical systems, it can perform various computations. Most implementations of logic gates or neural networks using a nonlinear chemical system require couplings between individual chemical reacting systems (such as a continuous-flow stirred tank reactor (CSTR) or a batch reactor) to compose networks. The couplings between units are experimentally accomplished by mass transfer,⁴ flow rate control,⁵ or electrical coupling.^9,10 Input data or patterns are encoded by a variation in flow rate or concentrations of chemicals and are processed through the network by an interaction between units coupled with each other in the implementations. Most chemical neural networks are implementations of classical neural networks based on the McCulloch–Pitts neuron model, in which information is assumed to be encoded in the form of a firing rate of neurons. In recent years there has been increasing interest in the spiking neuron, in which analog information is temporally encoded by the timing of action potential.¹¹ Many studies have suggested that information may be transmitted and processed by temporal coding in a nervous system. Temporal coding gives a more efficient mechanism for neural information processing than does firing rate coding, and offers an interpretation of the fast neurobiological information processing, such as visual and auditory perception in vertebrates, that cannot be described clearly by the firing rate coding. It has been shown by Hohmann et al.¹² that the temporal encoding of analog information is possible by using a nonlinear chemical system with threshold properties. They demonstrated Hopfield's neuron model¹³ of time-advance coding by the minimal bromate (MB) system with a subcritical Hopf bifurcation. In their demonstration, the MB system shows oscillation when a modulated flow rate periodically crosses the threshold periodically. By shifting the average flow rate by a bias, the phase of oscillations shifted by an amount in proportion to the bias is obtained, so the information included in the bias can be temporally encoded. Maass¹⁴ introduced a spiking neural network (SNN), which is composed of spiking neurons based on the temporal coding of analog information by using the timing of action potential. The SNN can perform analog computation in natural and efficient ways. Information processing in the SNN can be described briefly as follows: First, analog information is encoded by using the firing time of action potentials; that is, presynaptic neurons produce action potentials with a time advance in proportion to the size of input variables. Then a postsynaptic neuron receives the weighted sum of the post synaptic potentials (PSPs) and fires as soon as the sum of the PSPs crosses over a threshold of the postsynaptic neuron. Because information is represented and processed by using the time differences between spikes, a SNN has important advantages over classical neural networks, based on the firing-rate encoding and computing, in its computing power and biological reality. In this work, we demonstrate a chemical spiking neuron network (CSNN) theoretically on the basis of the formulation of SNN by using excitable chemical systems. The CSNN is a network composed of nonlinear chemical reacting systems which can perform the functions of spiking neurons. An excitable chemical system exhibits excitation phenomena very similar to those of neurobiological neurons: The system responds with a large amplitude spike (corresponding to PSP in neurobiological neuron) under a perturbation over the threshold.¹⁵ It will be shown that an excitable chemical system is able to perform the functions of a spiking neuron in SNN. Input variables are expressed and computed through networks as firing times. The fact that time is used as a resource of computations distinguishes the CSNN from other chemical networks. Information processing in CSNN is performed efficiently and naturally prior to those of other chemical neural networks because the CSNN possesses most of the important beneficial features of SNN.

The Oregonator¹⁶ is chosen as a theoretical model for an excitable chemical system in a CSTR. The Oregonator model produces a very large amplitude spike in a concentration of intermediates in the proper conditions when a perturbation on the concentration of an intermediate over the threshold is applied.^16b In a CSNN, each chemical neuron is required to be coupled appropriately. In our construction of a CSNN, a variable that triggers the excitation is selectively coupled with a corresponding variable in other systems by linear diffusion. Moreover, the linear diffusion is assumed to be unidirectional.

2 Excitable chemical system as a spiking neuron

In order to investigate whether an excitable chemical system is able to perform the functions of a spiking neuron, the quantitative relations between inputs from externals and an output of the system must be ascertained. The excitable chemical system as a chemical spiking neuron might receive inputs in the form of a change in concentrations of chemicals. The change will trigger a fire, or excitation, to give a large amplitude spike in the concentrations of the intermediates. The firing time is the output of the chemical spiking neurons. Maass' formulations are based upon rather simple spiking neuron models, which fire as soon as the sum of PSPs exceeds an excitation threshold. However chemical excitable systems do not show the same excitation mechanism as the spiking neuron model in Maass's formulation. The excitation phenomena of chemical excitable systems are not as simple as the simple spiking neuron model when the target measure becomes the firing time of excitation spikes. The dependences of the firing time on the inputs are considered in order to discover whether using chemical excitable systems as the chemical spiking neurons is possible. The four-dimensional Oregonator^16b is used as a theoretical model for the chemical excitable system. The reaction scheme with rate constants and other parameters is listed in Table 1. The Oregonator model generates a large amplitude spike of concentrations of Br⁻ and Ce³⁺/Ce⁴⁺ couples when perturbations as inputs are applied to concentrations of Br⁻.^15e The balance equations for the Oregonator with a perturbation as an input can be written as follows:

Table 1 Amplified Oregonator model^a

a All concentrations in M and time in s. Values of parameters are the same as in ref. 15e.
A + Y → X + P	ν 1 = k1[A][Y]
X + Y → 2P	ν 2 = k2[X][Y]
A + X + C → 2X + Z	ν 3 = k3[A][X][C]
2X → A + P	ν 4 = k4[X]^2
P → L	ν 5 = k5[P][Y]
P → Y	ν 6 = k6[Z]^1/2[P]
Z → C	ν 7 = k7[Z]
X≡HBrO2	A≡BrO3	k 1 = 2	k 5 = 1 × 10^5	[X]0 = 0
Y≡Br^−	L≡RBr	k 2 = 2 × 10^9	k 6 = 2 × 10^2	[Y]0 = 1 × 10^−6
Z≡2M^(n+1)+	C≡2M^(n)+	k 3 = 1 × 10^4	k 7 = 1	[Z]0 = 1 × 10^−3
P≡HOBr	k 0 = 0.0005 s^−1	k 4 = 5 × 10^7	[A] = 0.1	[P]0 = 0

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where

and x, y, z and p correspond to the concentrations of HBrO₂, Br⁻, Ce⁴⁺ and HOBr, respectively. The R_x, R_y, R_z and R_p represent the corresponding reaction rate terms and w is a weight. The system maintains stationary states (x_s, y_s, z_s, p_s) initially. The time-integrated f can be thought of as the PSP from other neurons and is assumed to be transported to the neuron at a constant time T_in in our CSNN model. All functions have a non-zero value only in the given time range, and zero value elsewhere. In order to investigate the dependence of the firing time on the PSP functions, eqns. (1)–(4) are solved numerically for the four different PSP functions listed in Table 2 and the results are shown in Fig. 1 and 2. When PSP with weight w and duration l is applied to the system at the time T_in, sharp spikes with large amplitude in y and z appear (Fig. 1). The firing times are obtained with variations of weight w (Fig. 2). The dependence of the firing time on the shapes of the PSP functions can be elucidated by investigating the firing time as a function of 1/w. The coefficient δ′ of the functions is adjusted to give almost the same range of 1/w for comparison between the four functions. The firing time, t_ν, is set, for convenience, to be the time when y − y_s exceeds 1 × 10⁻⁶ because the spike in y has a large and sharp peak in comparison with the other three variables. The firing time shows an almost linear increase in the range of low values of 1/w for all PSP functions. The results do not show any important dependence of the firing time on the shapes of the PSP functions, for a lower value of 1/w at least. From the results the firing time can be approximated for all cases by the following relation,

Table 2 Four elementary PSP functions

Fig. 1 A spike generation by the Oregonator. The arrow indicates the time when the perturbation is applied.

Fig. 2 The firing time obtained for four basic functions and spike of the Oregonator as PSPs. (a) Shapes of four functions. (b) The firing time increases linearly in the range [1/w_max, 1/w_min] = [1/100, 1/400] for all functions (s: spike of the Oregonator).

where t_ν is the firing time of the chemical neuron, Θ the slope of the curve, d the delay. Eqn. (5) gives a simple quantitative relation between the firing time and weight that is independent of the shape of the PSP functions. Θ in eqn. (5) is not a threshold of the Oregonator, but a coefficient intrinsic to PSP functions. The firing time from eqn. (5) has the same form as that in Maass' formulation except for its restriction on the range of the weight and the meanings of Θ and d. In Maass' model, the delay represents the time taken for the transportation of the PSP from a presynaptic to a postsynaptic neuron and is independent of the PSP. In our model there is no delay between the pre- and postsynaptic neurons, and the PSPs are transported to postsynaptic neurons instantly. Instead, the excitable chemical system gives a spike with delay, d, after the PSP exceeds the threshold.

For the use of the spikes in the Oregonator as the PSP in eqn. (5), another chemical system (which corresponds to a presynaptic spiking neuron) needs to be included for the construction of a monosynaptic system. The balance equations for y in the system can be written as follows:

Both systems also maintain stationary states, (x_s, y_s, z_s , p_s) initially. Perturbations are applied to the presynaptic chemical neuron in zⁱⁿ at time T_in^′, with amplitude δ′, and duration l. Then a presynaptic chemical neuron is excited and gives a spike at time T_in. The spike is transported to the postsynaptic chemical neuron instantly. Because a negative perturbation has to be applied in y, changing the sign of the perturbation to postsynaptic chemical neurons (see the sign of the perturbation term in eqn. (2)) is necessary. Eqns. (6) and (7) are numerically solved and the results are represented in Fig. 2 and 3. In the linear range of 1/w, the firing time of the spike of the Oregonator shows an excellent linear fit to eqn. (5). It is confirmed by a good fit of the firing time of the spike in y to eqn. (5) in Fig. 3. On the basis of this result we conclude that the Oregonator can be used as a model for chemical spiking neurons which comprise a CSNN by being coupled with each other.

Fig. 3 Dependence of firing time in monosynaptic system on 1/w. Dots represent numerical results of eqn. (2) and the solid line is a linear function fitted by eqn. (5).

It is worth noting that the present work is not intended to suggest the excitable chemical system as a neurobiological neuron model. There exist more realistic models like the classical Hodgkin–Huxley model.¹⁷ The Oregonator cannot reflect features of the action potential of the neuron (Fig. 1) thoroughly, and couplings between the neurons is not accomplished by unidirectional diffusion as used in CSNN.

3 Chemical spiking neuron network

In SNN, input variables are encoded and processed by a firing time of spikes through networks. Spikes of spiking neurons are delivered to other neurons in the next layer and the weighted sum of the spikes excites the neurons again. Output values are obtained by the timing of the spikes of output neurons as a result of the processing. From the results, a single chemical spiking neuron CSNN will be constructed. To describe how to construct the CSNN for a given SNN, a simple single layer SNN will be used as an example. The single layer SNN may be composed of n input neurons that accept the input pattern vector (s₀, s₁, s₂, ..., s_n) encoded by the timing of input spikes and of one output neuron that receives the weighted sum of PSPs from input neurons and exhibits spikes delayed by the amount of output. For each reactor, the balance equations for y can be written as follows:

for input neurons

and for one output neuron

All the neurons in the network are maintained in stationary states initially. The input neurons u_i (i = 0, 1, 2, ..., n) are perturbed at time T_in^′ − s_i and fire at the time T_in − s_i (all input neurons have the same delay after the threshold is exceeded because the same δ′ is used for all input neurons). The resulting spikes are instantly transmitted to the output neuron. When the weighted sum of PSPs or the size of perturbations in y^out exceeds the threshold of chemical neurons, the output neuron, v, fire after the delay, d. A PSP of an input neuron corresponds to a time-integrated spike in z_iⁱⁿ. Eqn. (5) can be rewritten as follows,

For the CSNN it is assumed that

provided that

and

where Δ is the time width of the spike of the chemical neuron (PSP). According to Maass' formulation, eqn. (11) is equiva lent to

with

where r_i = w_i/λ and the fixed value s₀ = 0 is used. The w₀ is chosen so that ∑_{i = 0}ⁿw_i = λ. Eqn. (13) becomes

All parameters except for γ can be obtained from the results for the monosynaptic system and eqn. (16) can be satisfied by appropriately setting γ. In the CSNN the additional condition of eqn. (12) must be satisfied for the CSNN to function as a SNN appropriately. By confining λ⩽w_max, the condition ∑_{i = 0}ⁿw_i⩽w_max can be satisfied, but because all input neurons do not fire necessarily at the same time, an auxiliary neuron needs to be included in the network to guarantee ∑_{i = 0}ⁿw_i⩾w_min. We use the nth neuron as the auxiliary neuron by setting w_n = w_min and s_n = γ. Then eqn. (14) can be rewritten as follows:

with

A simple chemical network composed of six input neurons and one output neuron is constructed and simulated numerically to investigate whether the chemical neuron network satisfies eqn. (17). The weight and signal of five input neurons (except for auxiliary neuron) are randomly determined and the firing time of the output neuron is obtained. The numerical results shown in Fig. 4 show good agreement with the firing time given by eqn. (17).

Fig. 4 The firing times of the output neuron in the simple CSNN obtained for 100 randomly chosen weight vectors. The solid line represents the firing time given by eqn. (17) (T_out = 50, γ = 0.3, λ = 5.6 × 10⁻³).

4 Discussion

In this work we have demonstrated a CSNN theoretically by using coupled excitable chemical systems. We have attempted to examine the functionalities of excitable chemical systems as spiking neurons for the construction of a CSNN. It is shown that the Oregonator can be used as a model for the chemical spiking neuron, able to function as a unit of a CSNN. Chemical spiking neurons have excitation mechanisms different from the spiking neuron, but the chemical spiking neuron mimics the behavior of the spiking neuron with its restriction of the weights of couplings between chemical spiking neurons. There seems to be no important dependence of firing time on the shape of the PSPs, and the firing time can be expressed as an inverse linear function of weight for four elementary functions and a spike of the Oregonator. The fact that there is no obvious dependence of the firing time on the shapes of PSP is an important feature of the chemical spiking neuron. It offers a simple model for processing temporally coded information and can be replaced by other chemical systems that show excitability. The CSNN can be constructed by coupling the chemical spiking neurons [italic v]ia unidirectional linear diffusion. In our model of the CSNN, an auxiliary chemical neuron is required for appropriate functions. If couplings between chemical systems in the experimental system for the CSNN are accomplished by using electrical current, the CSNN can process information faster than other computing networks that are composed of nonlinear chemical systems coupled with each other through parameters such as flow rates. Moreover, the CSNN has all the merits of a SNN in its computing performance. We hope that this work reveals the prospects for the new use of aspects of nonlinear chemical systems as computational devices and gives some direction for the further understanding of information processing in biological systems.

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