An array of nonidentical and locally connected chaotic biological neurons is modelled by a single representative chaotic neuron model based on an extension of the Hindmarsh-Rose neuron. This model is then employed in conjunction with the unscented Kalman filter to study the associated state estimation problem. The archetypal system, which was deliberately chosen to be chaotic, was corrupted with noise. The influence of noise seemed to annihilate the chaotic behaviour. Consequently it was observed that the filter performs quite well in reconstructing the states of the system although the introduction of relatively low noise had a profound effect on the system. Neither the noise-corrupted process model nor the filter gave any indications of chaos. We believe that this behaviour can be generalised and expect that unscented Kalman filtering of the states of a biological neuron is completely feasible even when the uncorrupted process model exhibits chaos. Finally the methodology of the unscented Kalman filter is applied to filter a typical simulated ECG signal using a synthetic model-based approach.

Oscillatory signals in the cardiovascular region either originate directly from the sinoatrial node or one of the neurons as an action potential traverses to the ventricle myocytes. Alternatively they are functions or weighted sums of action potentials arising at spatially distributed points. To consider a range of oscillatory measurements in the cardiovascular region, it is important to consider the output of typical neuronal cell.

Neural information is mainly encoded in various firing patterns of a neuron, such as periodic spiking (or bursting) and chaotic spiking (or bursting), travelling among coupled neurons within a physiological domain of neurons such as the heart. The “action potential” is a spontaneously and rhythmically produced electrical impulse in a membrane of neuron cell that occurs during the firing of the neuron due to an exchange of charged ions inside and outside a neural cell. Although not a definition, a dynamic system may be considered chaotic if it exhibits (i) sensitive dependence on the initial conditions and (ii) a number of dense orbits with a multiplicity of periods for a range of parameters. Two nonlinear dynamic systems with chaotic responses can sometimes exhibit the phenomenon of synchronization when the responses of the two lock in and seem to drive each other with a common feature such as the phase, phase-lag, amplitude, and envelope or even some generalised property that can be described in terms of a functional of the features of the response. Physiological observations have confirmed the existence of synchronous motion of neurons in different areas of the heart (Elson et al. [

In this paper the state and parameter estimation of an array of nonidentical, locally connected chaotic biological neuronal models is considered. It is known that, under certain conditions, even a single biological neuron can exhibit chaotic behaviour. Chaos may be achieved by introducing the nonlinear effects of the chemical and electrical synapses. Alternately, the chaotic behaviour of the single biological neuron is achieved by driving it with periodic excitations. The global behaviour of an array of biological neurons may then be investigated by considering a spatial distribution of identical neurons, where spatiotemporal chaos emerges, as well as in presence of spatial diversity, generated by a distribution law which could be stochastic or chaotic. In the latter case, it has been observed that the introduction of spatial disorder enhances the self-organization or synchronisation capability. In particular, in agreement with the results presented in the works of Elson et al. [

In this paper, we consider a typical extended four-state Hindmarsh-Rose (HR) model (Hindmarsh and Rose, [

Finally the methodology is applied to the Electro-cardiogram (ECG) measurements which are modelled as oscillatory signals using a synthetic model first proposed by McSharry et al. [

The analysis of biological neurons had that shown they could be modelled with only three or four states, we chose initially to use a familiar simplified model put forward by Hindmarsh and Rose [

Adding the term

The neuron model described by (

(a) Extended HR neuron: state-response plot comparing response of three states to input. (b) Extended HR neuron: close-up of state-response plot of the first two states illustrating chaos. (c) Extended HR neuron: close-up of the Poincaré map of the chaotic first state corresponding to Figure

Most dynamic models employed for purposes of estimation neural action potential signals are generally not linear. To extend and overcome the limitations of linear models, a number of approaches such as the extended Kalman filter (EKF) have been proposed in the literature for nonlinear estimation using a variety of approaches. Unlike the Kalman filter, the EKF may diverge, if the consecutive linearizations are not a good approximation of the linear model over the entire uncertainty domain. Yet the EKF provides a simple and practical approach to dealing with essential nonlinear dynamics.

The main difficulty in applying the EKF algorithm to problems related to the estimation of a neural action potential signal is in determining the proper Jacobian matrices. The UKF is a feasible alternative that has been proposed to overcome this difficulty, by Julier et al. [

The UKF gets its name from the unscented transformation, which is a method of calculating the mean and covariance of a random variable undergoing nonlinear transformation

Consider a random variable

Given a general discrete nonlinear dynamic system in the form

The UKF is based on approximating the probability distribution function than on approximating a nonlinear function as in the case of EKF. The state distributions are approximated by a Gaussian probability density, which is represented by a set of deterministically chosen sample points. The nonlinear filtering using the Gaussian representation of the posterior probability density via a set of deterministically chosen sample points is the basis for the UKF. It is based on statistical linearization of the state dynamics rather than analytical linearization (as in the EKF). The statistical linearization is performed by employing linear regression using a set of regression (sample) points. The sigma points are chosen as the regression points. The mean and covariance at the sigma points then represent the true mean and covariance of the random variable with the particular Gaussian probability density. Thus when transformed to the nonlinear systems, they represent the true mean and covariance accurately only to the second order of the nonlinearity. Thus this can be a severe limitation of the UKF unless the nonlinearities can be limited to the first and second order in the process model.

The success of the application of the UKF depends largely on the approximation to the covariance which is estimated as a weighted linear sum of the covariance at the sigma points. When this approximation is such that the covariance is not positive definite, the UKF algorithm fails as the Cholesky decomposition is not possible. To ensure that this covariance is essential, adjust the scaling parameter

Neuron model states: plots of the simulated and estimated states on the same scale versus the time in seconds.

Figure

Neuron model states: plots of the UKF estimate errors in the state variables versus the time in seconds.

Neuron model states: plot of the evolution of estimated measurement error versus the time in seconds.

In particular we observe the relatively large error in the third state,

McSharry et al. [

A typical

The original model proposed by McSharry et al. [

Parameters of the ECG model given by (

Index ( | 1 ( | 2 ( | 3 ( | 4 ( | 5 ( |
---|---|---|---|---|---|

Time (secs) | |||||

0 | |||||

As rightly pointed by Sameni et al. [

The

Typical initial conditions for the states in (

1.5 | 50.0 | ||||

34.7222 | |||||

0 | 55.4017 | ||||

78.1250 | |||||

8.0 |

Given a set of continuously sampled ECG measurements, the measurements may be expressed by the equation

In Figure

Comparison of simulated and estimated responses of the states

Errors in the estimated responses of the states

In Figure

Measurement estimate error, plotted against the number of time steps.

The unscented Kalman filtering method was applied to estimate the states of an HR-like neuron model which in the absence of noise were deliberately chosen to be chaotic. The process and measurement was then corrupted by the introduction of very low levels of white noise. The noise had a profound effect on the response of the model as it seemed to annihilate the chaos. It was observed that the filter performs quite well in reconstructing the states of the system. Neither the filter nor the noise corrupted process model gave any indications of chaos. Moreover, the exercise gave us valuable experience in applying the UKF to a biological neuron. Preliminary studies of the application of the UKF to a Hodgkin-Huxley type model indicated that the successful application of the unscented approach to an ensemble of biological neurons was feasible, provided the sigma points were scaled according to certain scaling laws related to the gate constants. Finally the methodology of the unscented Kalman filter is successfully applied to filter a typical simulated ECG signal using a synthetic model-based approach.