This paper simulates an analogue self-learning pulse neural network based on memristive elements, taking into account their stochastic properties. A variable-resistor model of a thinfilm memristor based on an exponential model for dopant drift is used as a memristor model. Stochastic properties are accounted for by a term in the memristor equation of state, which presents an additive (Gaussian) noise. Memristor switch from low-resistance to highresistance state in this case occurs differently from cycle to cycle, corresponding to the experimental data. The mathematical model previously developed by the authors is used; it describes the analogue implementation of a pulse self-learning neural network memristive elements as synaptic weights and a learning mechanism based on the STDP method. The operation of a network consisting of five neurons with 320 synapses for recognition of various black and white images is simulated. As a result of the simulation, the network was successfully learnt to recognize the given patterns VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),
Artificial neural networks play a great role in modern life [
One of the factors restraining the neural network development is the high computational
complexity of the corresponding neural network algorithms: a network training time can be measured
in weeks and months. Currently, to speed up their work, research is being conducted on creating
special processors [
In conjunction
with the development of specialized computing devices, the use of other
computational principles seems promising, namely analogue computations instead of digital ones,
since they are performed in orders of magnitude faster. Relatively new electrical elements
memristors [
2021 Copyright for this paper by its authors.
it keeps its conductivity, which allows it to be used as an elementary memory cell, and it is possible to
dynamically change its resistance in the presence of current. This is a certain similarity of the
memristors’ properties with the properties of biological synapses, allowing them to be used to create
analogue neural networks [
Basically, the memristive effect occurs in various oxides due to the movement of ions (oxygen
vacancies) and the formation/destruction of conducting filaments. The ion movement is random and,
as a consequence, memristors have certain stochastic properties. A detailed experimental and
theoretical study of this effect was carried out in [
Mathematical models of memristors are traditionally formulated as dynamic systems with respect
to the parameter of memristor state, which characterizes the level of element’s conductivity [
In this work we simulate a two-layer full-connected network with one layer of memristor elements
(synapses) [
The main purpose of this scientific work is to simulate the functioning of the self-learning pulse neural network with the hardware implementation based on memristive elements, taking into account their stochastic properties in learning the recognition of five images.
In the second section, a model of a variable-resistor thin-film memristor, based on an exponential model of dopant drift, is presented. The third section provides a mathematical model of the neural network. Next, the neural network operation is simulated and, finally, the main results of the work are formulated.
An approach based on representation of a memristor as a dynamic system with a generalized state
variable is widely used for simulating the memristor’s operation [
Where F - rate of change of the state variable; - coefficient characterizing noise intensity; W Wiener process.
Concretization of the functional dependence F ( x) and relation of the state variable with physical parameters gives a memristor model. In the present paper, a variable-resistor thin-film memristor model based on the exponential model for a doping impurity drift is used [19]: V v Dp2 exp RVopn I dt , V V p , dx = dW + v DVn2 exp RVonn I dt , V Vn , v RDo2n Idt , Vn V V p , R = Ron x + Roff (1 − x),
V R I = where x 0, 1 — state variable; Ron, Roff — minimum and maximum memristor resistance; I , V , R — actual memristor’s current, voltage, and resistance values; Vp , Vn — values of voltages at which state switching occurs; v —doping mobility coefficient; D —semiconductor film thickness; — coefficient characterizing the noise intensity; W — Wiener process.
To obtain an approximate realization of the stochastic process x (t ) we use the Runge-Kutt method of order 1.5. We determine an approximate solution using the grid t0 t1 t2 ... tN −1 tN under the initial condition x(t0 ) = x0 2
1 xk +1 = xk + 4 3F ( x1k ) + F ( xk2 ) tk +1 + Wk +1, where tk +1 = tk +1 − tk, Wk +1 = W (tk +1 ) − W (tk ). Let N (0,1) is a normally distributed random variable with zero mathematical expectation and unit variance. Then the random value Wk +1 is calculated as Wk +1 = zk +1 tk +1 , where zk +1 is chosen from N (0,1).
Note that we can use other methods of integrating stochastic ODEs, such as the Milstein method of first order, which for this problem would be equivalent to the Euler-Maruyama method. In this case, the solution on the (k + 1)-th time layer will be defined as xk +1 = xk + F ( xk ) tk +1 + Wk +1.
In this work, the memristor’s operation was simulated the following parameter values: Ron = 205 Ohm, Roff = 2128 Ohm, v = 6 10 −10, V p = 0.65 V, Vn = −0.87 V, D = 621 nm, x(0) = 0, V (t ) - see at
The presence of noise in the memristor model causes all parameters to take on the stochastic properties. On the diagram with the volt-ampere characteristic, we can distinguish trajectories that correspond to different switching cycles of the memristor. Here we observe the good correspondence in the right part of the diagrams and satisfactory correspondence in their left parts.
We consider a two-layer fully connected self-learning analogue network with one layer of memristor elements (synapses); it consists of 64 inputs and 5 neurons (Fig. 3).
According to the STDP method, the learning mechanism is implemented through feedback (Vte). At the moment of neuron activation, two pulses of opposite sign arrive via the feedback channel with delays. If there is activity at the synapse and a positive feedback pulse arrives, then the resistance value of the corresponding memristor decreases, and if a negative feedback pulse arrives, the memristor resistance increases.
The neuron model is a parallel RC circuit. As soon as the value of the potential across the capacitor exceeds a certain threshold, its potential is reset, and signals Vte and Vout are generated. In addition, at the moment of neuron activation, the potential of other neurons decreases in proportion to the coefficient .
The mathematical model is set in accordance with [
FX ( xi, j , Vtej − Vinjt ) dt + dWi, j , Vgi (t ) 0, dxi, j = 0, Vgi (t ) = 0, V v Dp2 exp RVopn RVi, j dt, V V p , FX ( xi, j ,V ) = v Dn2 exp RVonn RVi, j dt, V Vn , V
dt, Vn V V p ,
Ron V v D 2 Ri, j Ri, j = Ron xi, j + Roff (1 − xi, j ),
dVinjt = 1 n dt Cint Vgi (it=)1 0 Ri, j d j = 1 − (Vinjt − Vth ) j, dt
Vo+ut , j out , Voujt = 0, out j , Vte+ , j s , Vte− , r j r + s ,
2 2 Vtej = Vte0 , r j ,
Vˆtej − Vinjt − Vinjt − max (Vinit − Vth )ˆi, j m (Vinit − Vth ) Vinjt ,
Rint i=1,m i=1
0, s j r r + s j r , 2 2 ˆi, j = 1 − (1 − ij ), Vˆtej = max(0, min(Vtej , Vte0 )), xi, j (0) = random 0,1, Vinjt (0) = 0, j (0) max( r , out ), i = 1...64, j = 1...5, where V gi is the actual voltage value at the i-th input of the neural network; Vtej - the actual voltage value in the feedback of the j-th neuron; Voujt - the actual voltage value at the output of the j-th neuron; j - time elapsed since the last activation of the j-th neuron; Vinjt - voltage across the capacitor of the jth neuron; Rint , Cint - the value of resistance and capacitance of neurons; Vte+, Vte−, Vte0 - the amplitude values of the feedback pulses and the default voltage value; Vo+ut - the output pulse amplitude; Vth - the neuron activation voltage level; Ri, j - the memristor’s resistance value of the i-th synapse of the j-th neuron; xi, j - the memristor’s state of the i-th synapse of the j-th neuron; r - the feedback signal duration after neuron activation; s - the duration of one pulse in the feedback signal, 2 s r ; out the duration of one pulse at the network output; α - suppression coefficient; ij - the Kronecker symbol; ( x) - delta function; ( x)- Heaviside function; - coefficient characterizing the noise intensity; Wi, j- Wiener process corresponding to the i-th memristor of the j-th neuron.
The recognition problem for five patterns is considered (Fig. 4) [21]. The process of training the network is as follows: for each epoch (equal to r 2 seconds), the input signals vector Vg (t ) corresponds either to one of the recognized patterns or is set randomly (the elements of the vector 0.27). Over time, the network adapts to pattern recognition. The Distribution of patterns among neurons occurs in the course of training.
The parameters of the mathematical model of the neural network are adjusted depending on the memristor model. For the used model, which corresponds to a memristor based on titanium oxide (TiO2), we have the following parameter values: Rint = 200 Ohms, Cint = 45 μF, Vte+ = 0.7 V, Vte− = −0.9 V, Vte0 = 10 mV, Vo+ut = 2 V, Vth = 9 mV, r = 3 ms, s = 50 μs, out = 1.5 ms.
Fig. 5 shows the process of synaptic weight adaptation to recognized patterns. The shade of grey corresponds to the state variable value of the corresponding memristor: the darker, the greater the conductivity; the lighter, the less. At the initial moment of time, all weights are initialized with random values, and gradually change during the network operation. From about the 1200th era, patterns began to be seen, the recognition of which was trained by the network: the information was memorized by the neural network. The duration of one epoch is 1.5 ms. The correspondence of the network weights to the patterns indicates that the network has successfully trained to recognize the given images. Due to the stochastic component in the memristor model, patterns can be distributed among neurons in different ways and the adaptation of weights can occur at different rates.
The work is devoted to simulating the operation of an analogue self-learning pulse neural network built on the basis of memristive elements in the image recognition mode. The simulation is carried out taking into account the stochastic properties of memristors. A variable-resistor thin-film memristor model based on an exponential dopant drift model, in which there is a term responsible for additive (Gaussian) noise, is considered. The ampere-voltage characteristics of the model are compared with experimental data on titanium oxide. The simulating of the operation for the network consisting of five neurons with 320 synapses, designed to recognize five different black-and-white images of 8 by 8 pixels, is performed. As a result the network has successfully trained to recognize the given patterns.
The work was performed with the support of RFBR grant No. 19-29-03051mk.
The studies were carried out using the resources of the Center for Shared Use of Scientific Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the Russian Academy of Sciences", funded by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation under project No. 075-15-2021-663.
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