Nonlinear Dynamical Systems with Chaos and Big Data: A Case Study of Epileptic Seizure Prediction and Control

Abstract

The modeling of dynamic behavior of systems is a ubiquitous problem in all facets of human endeavors. Importantly so, dynamical systems have been studied and modeled since the nineteenth century and currently applied in almost all branches of sciences and engineering including social sciences. The development of computers and scientific/numerical methods has accelerated the pace of new developments in modeling both linear and nonlinear dynamical systems. However, modeling complex physical system behaviors as nonlinear dynamical systems is still difficult and challenging. General approaches to solving such systems typically fail and require personalized problem dependent techniques to satisfy the constraints imposed based on the initial conditions to predict state space trajectories. In addition, they require enormous computational power available on supercomputers. Numerical tools such as HPCmatlab enable rapid prototyping of algorithms for large scale computations and data analysis. BigData applications are computationally intensive and I/O bound. An example, state of the art case study involving big data of epileptic seizure prediction and control is presented. The nonlinear dynamical model is based on the biology of the brain and its neurons, chaotic systems, nonlinear signal processing, and feedback and adaptive systems. The goal is to develop new feedback controllers for the suppression of epileptic seizures based on electroencephalographic (EEG) data by altering the brain dynamics through the use of electrical stimulation. The research is expected to contribute to new modes of treatment for epilepsy and other dynamical brain disorders.

Appendices

Appendix 1: Electrical Stimulation and Experiment Setup

Deep Brain Stimulation (DBS) protocols in several “animal models” of epilepsy have shown some effectiveness in controlling epileptic seizures with high frequency stimulation targeting the subthalamic nucleus, anterior thalamic nucleus, caudal superior colliculus, substantia nigra, and hippocampus (Vercueil et al. 1998; Lado et al. 2003). All these investigators used stimulation parameters in the following ranges: frequency from 50–230 Hz, bipolar constant current pulses (30–1000 μs) at current intensities from 0.1 to 2 mA. In contrast, low frequency (between 1 and 30 Hz) stimulation resulted in increase of seizure susceptibility or synchronization of EEG. The electrical stimulation we used in our experiments had a pulse width of 300 μs at a current intensity ranging from 100–750 μA and pulse-train duration of 10–20 min. Considering possible induced tissue damage by electrical stimulus, the maximum current intensity of 750 μA falls under the safe allowable charge density limit of 30 μC/cm² as reported by Kuncel and Grill (2004) for deep brain stimulation; given the size of the stimulating electrode and pulse parameters chosen. While these specific values of the stimulation parameters are being utilized in our work, their optimization can become a project on its own right and is not considered at this point.

The stimulation is applied between pairs of electrodes (amygdalar, hippocampal, thalamic, frontal) according to the localization analysis results, whenever a seizure warning is issued by our seizure warning algorithm or in the case of the offline stimulation in an open-loop manner for a fixed duration on various pairs of electrodes. A stimulation switching circuitry was developed in-house for the purpose of stimulating two sites at will. This equipment consisted of an Arduino Mega and printed circuit board containing electronic switches. Figure 13.15 shows a sketch of hardware setup used for all experiments.

Fig. 13.15

Block diagram showing the experimental setup used. The Switching circuit is controlled by the Arduino Due which in turn gets commands from the MATLAB program running on the computer. The switching circuit enables us to stimulate any pair of electrode at will. The stimulation signal is generated from the A-M systems stimulator

The Intan RHD2000 development board and its 16 channel amplifier was setup as the EEG acquisition machine. The rats used in our study have 10 EEG channels, a ground and a reference channel. We collect EEG data from 10 electrodes located in different parts of the rat brain (see Fig. 13.16). The EEG channels go through a switch board into the Intan EEG data acquisition system. The Intan board has an amplifier that conditions the signal so that its 16 bit ADC has sufficient resolution in the EEG waveforms which are typically in the 100s of μV range. Once digitized, the data is collected in an FPGA buffer until a MATLAB program polls it from the buffer over USB. The MATLAB code operates every 2 s and it brings in 2 s worth of data that are sampled from the ADC at 512 Hz from all channels. On MATLAB, either the offline fixed stimulation code or the seizure warning algorithm decide on when and how to stimulate and those parameters are sent over emulated Serial Port (USB Virtual COM port) to an Arduino MEGA. The MEGA then commands the stimulator to provide stimulation on its output port. The amplitudes are fixed using analog knobs and are not programmable. Another function of the Arduino is to command the switch board so that it disconnects a chosen pair of electrodes from the rat to the EEG board and connects them to the stimulator so that the stimulation signal can pass through to the rat brain. Once stimulation needs to be switched off, the Arduino commands the stimulator to switch off and reconnects the EEG channels to the rat electrodes.

Fig. 13.16

Diagram of surgical placement of electrodes in the rat’s brain (top view)

Appendix 2: Preparation of Animals

The animals used in this study were male Spraque Dawley rats, weighing between 200–225 g, from Harlan Laboratories. All animal experimentation used in the study were performed in the Laboratory For Translational Epilepsy Research at Barrow Neurological Institute (BNI) upon approval by the Institutional Animal Care and Use Committee (IACUC). The protocol for inducing chronic epilepsy was described previously by Walton and Treiman (1988). This procedure generates generalized convulsive status epilepticus (SE). Status epilepticus was induced by intraperitoneal (IP) injection of lithium chloride (3 mmol/kg) followed by subcutaneous (SC) injection of pilocarpine (30 mg/kg) 20–24 h later. Following injection of pilocarpine, the EEG of each rat were monitored visually for clinical signs of SE noted behaviorally by the presence of a Racine level 5 seizure (rearing with loss of balance, Racine 1972). At EEG Stage V (approximately 4 h after pilocarpine injection) SE was stopped using a standard cocktail of diazepam 10 mg/kg, and Phenobarbital 25 mg/kg, both IP. The rats were then kept under visual observation for 72 h within which all measures were taken to stop them from deceasing. In the event that none of the methods to keep them alive worked, the animals were euthanized.

After SE was successfully induced in the animals, they were allowed 5 weeks for the seizure frequency to stabilize. Following this 5 week period, the animals were taken into surgery and an electrode array, as shown in Fig. 13.16, were implanted into their brain. Not including the reference and ground connections, each rat had 10 electrodes implanted. After surgery, each animal was allowed a week before being connected to an EEG machine. The referential voltages from each of the 10 electrodes mentioned was then recorded using an EEG machine (Intan RHD2000 development board).