5.2 The synchronization of two FitzHugh-Nagumo neurons
5.2.1 The effect of noise reduction on the synchronization two FitzHugh-
A very important area of research for nonlinear dynamical systems is the effect of noise, since noise can drastically modify the dynamics of a system. It can be present as a driving force or as an unwanted disturbance causing nondeterministic, or sometimes
Figure 5.1: The tracking error (x1−y1) achieved by controllersC1 andC2: C1 without disturbances (upper left); C2 without disturbances (upper right); C1 with disturbances (lower left);C2with disturbances (lower right). RFPT reduces the error every time to its one third.
chaotic behavior. The noise can appear as a consequence of a physical/dynamical attribute, e.g. thermal noise in every system warmer than absolute zero or the noise in nerve cells coming from synaptic events. In this case, basic physical knowledge is needed to understand, and if it is necessary, avoid the effects [77]. The other possibility is the unavoidable measurement noise, i.e. when the responses or state variables of a system are not calculated, but measured by some sensors affected by unknown external forces.
When a sensor is used the user cannot assume that it measures everything correctly.
Sensors always involve measuring noise. Luckily, nowadays many solutions exist to filter this kind of noise (e.g [78, 79, 80, 81, 82, 83, 84]). In the present literature the great majority of noise filters follow the model-based approach. The greatest impact on the field is given by Rudolf K´alm´an who introduced the model-based state estimation in the early sixties [85, 86, 87, 88, 89].
5.2 The synchronization of two FitzHugh-Nagumo neurons
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Figure 5.2: The realized ¨y1 values vs. the desired ¨yDes1 values for controllers C1 and C2: C1without disturbances (upper left);C2 without disturbances (upper right);C1with disturbances (lower left); C2 with disturbances (lower right). In ideal case one single straight line could be seen. With controllerC2 the figure shows almost one straight line, but with controllerC1there is a significant difference between the system responses.
The use of Kalman filters are based on certain assumptions regarding the statistical nature of the occuring noises. Furthermore, the Kalman filters have to be designed very carefully to avoid divergences [90].
A big disadvantage of Kalman filters is that whenever there is no reliable system approximation they cannot be applied. A different approach, the application of model-independent filters become necessary. In this section, it is examined how the filtering affects the performance of Robust Fixed Point Transformations. For the illustration, first a simple noise filter, an approximate model, and a controller are constructed.
A linear noise filter can be modeled in two ways in the time domain: in continuous case as an integral function as ˜f(t) :=R∞
0 F(τ)f(t−τ)dτ or in discrete case as a sum f˜k := P∞
i=0Fifk−i. f denotes the systems’ states, F(τ) is a monotonously decreasing
function, and Fk-s are discrete weights that normally converge to zero as τ, k → ∞.
They correspond to the “forgetting speed” of the filter. They can be calculated in discrete case as Fk :=βk(1−β) with 0 < β <1. For saving space and time Fk-s can be calculated by a simple buffer P: Pn+1 = βPn+fk+1, ˜fk+1 = (1−β)Pk+1. The actual value ofβ directly influences the “memory” of the filter: the larger value β has, the longer memory the filter has. In the following, the approximate model and the controller are detailed.
In fact, the approximate model is an inverse model as explained in Chapter 4. For simulation purposes, the same parameter settings are used as in the previous section:
ˆ
a = 12, ˆb = 1.5, and ˆg = 0.5. By the use of the same PD controller relatively good tracking accuracy can be achieved if the feedback parameters are big enough.
However, big feedback in the derivative term causes big disturbances in case of noisy signals. Within the frames of the PD controllers reduced feedback coefficients result in inaccurate tracking. The real virtues of the RFPT-based controller can be well observed when a relatively fast signal has to be slowly approximated. On this reason, in this section the same type of controller is applied with Λ = 0.2fexct. For improving the controller the second RFPT-version is chosen (see Section 4.4). In the following, simulation details and results are presented.
The task is to synchronize two chaotic FitzHugh-Nagumo neurons described in Section 3.1. In (3.2) x1 and x2 denote the master system whiley1 and y2 describe the state of the slave system. Regarding the control task, assume thaty1 has to precisely track x1 by properly setting ˙u. For calculating the differentiated control force, the observation of ¨y1 and ¨x1 is needed. Assume furthermore that there are simple sensors directly observing x1 and y1. The sensors have some observation noise that spreads through the numerical derivation with finite time-resolution ∆tCycle. To reduce the effects of this noise the above detailed filter is used.
The simulations are made in Scilab-5.1.1 [76] (developed by the Consortium Scilab (DIGITEO)) and the related graphical programming tool SCICOS 4.2. For SCICOS the maximum step size of the integration is not limited in the simulations. In the illustrative examples the Integrator absolute tolerance parameter is set to 0.01, and the Integrator relative tolerance is 0.001. In the RFPT-based case the ∆tCycle = 0.1ms cycle time is chosen withK =−1000000,A= 5×10−7 andB = 1 settings. The initial values for the state variables arex1= 0, x2 = 0,y1 = 0.005, andy2 = 0.
5.2 The synchronization of two FitzHugh-Nagumo neurons
Since the adaptive controller works by observing ¨x1and ¨y1it may be sensitive to the observation noises. In this approach, in noisy case, it is assumed that the quantitiesx1 andy1are directly observed as noisy signals. Their 1stand 2ndderivatives are estimated as ˙x(tn)≈[x(tn)−x(tn−1)]/∆tCycle, ¨x(tn)≈[x(tn)−2x(tn−1) +x(tn−2)]/∆t2Cycle. The estimated values can be filtered accordingly. Theβ = 0 case corresponds to no filtering, while the 0< β <1 corresponds to some filtering with shorter or longer filter memory.
The simulations are made to be able to compare two controllers: without RFPT (C1) and with RFPT (C2), to be able to compare results without and with measurement noise, and to be able to compare two noise filters: filter with β = 0.9 (F1) and filter with β = 0.5 (F2). In this section, six cases are presented: 1. with controller C1, without noise, without filter; 2. with controller C2, without noise, without filter; 3.
with controllerC2, with noise, without filter; 4. with controllerC2, without noise, with filter F1; 5. with controller C2, with noise, with filter F1; 6. with controller C2, with noise, with filter F2.
The first group of figures (Fig. 5.3) reveals the tracking error of the first state variable of the two systems e=x1−x3 in all of the six cases. It reveals that without noise and filteringC2 can gain more smooth and a little bit better trajectory tracking.
With noise and without filter the initial error is very high (10000%) though later it converges to zero. The F1 (without noise) does not increase the error, moreover it damps it a little bit. In the lowest figures, the results are shown with noise and with filters F1 (β = 0.9) and F2 (β = 0.5). F1 gives significantly better results than the other filter.
The second group of figures (Fig. 5.4) reveals the connection of the desired and realized system responses ¨yDes1 −y¨1. The bow–tie shaped figure in the first chart reveals that withC1(due to the parameter estimation errors) the trajectory tracking is not precisely realized. The effects of RFPT can be well observed in the second figure:
¨
y1 becomes almost identical to ¨y1Des. The last non chaotic figure (without noise but with filtering) does not differ from the second one which means that the filter does not “disturb” the improved results of RFPT. According to the simulations the noise filtering with β = 0.9 seems to work well. The other figures are noisy because the feedback in the derivative term of the PD controller causes disturbances in case of noisy signals.
Figure 5.3: Tracking error of the first state variable e1 = x1−y1. Upper left: with controllerC1, without noise, without filter; Upper right: with controllerC2, without noise, without filter; Middle left: with controllerC2, with noise, without filter; Middle right: with controllerC2, without noise, with filterF1; Lower left: with controllerC2, with noise, with filter F1; Lower right: with controller C2, with noise, with filter F2. The filter improves the results in every case and does not disturb RFPT.β = 0.9 (F1) seems to be the best choice.
5.2 The synchronization of two FitzHugh-Nagumo neurons
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Figure 5.4: Response error ¨y1Des−y¨1. Upper left: with controller C1, without noise, without filter; Upper right: with controllerC2, without noise, without filter; Middle left:
with controller C2, with noise, without filter; Middle right: with controller C2, without noise, with filterF1; Lower left: with controllerC2, with noise, with filterF1; Lower right:
with controllerC2, with noise, with filter F2. ControllerC2 brings the best results. The other cases are noisy because the noise is not filtered out in this level. β= 0.9 (F1) seems to be the best choice.