Bursting Oscillations of Neurons and Synchronization

Transcription

1 Bursting Oscillations of Neurons and Synchronization Milan Stork Applied Electronics and Telecommunications, Faculty of Electrical Engineering/RICE University of West Bohemia, CZ Univerzitni 8, 3064 Plzen CZECH REPUBLIC Abstract: - There are several motivations to be interested in fast-slow dynamics. For instance, many physiological or biological systems display different time scales. The bursting oscillations which can be observed in neurons, β-cells of the pancreas and population dynamics are studied and analyzed as fast-slow systems. This framework yields a natural generalization to the notion of bursting oscillations where, for instance, the active phase is chaotic and alternates with a quiescent phase. In the first part of the article, we introduce dynamics with bursting oscillations based on the Hindmarsh Rose system. Next part describe the bursting chaotic synchronization of two coupled neurons is simulated since the synchronization of individual neurons is thought to play a key role in Parkinson s disease, essential tremor, and epilepsies. On the end discrete version of bursting oscillators is introduced and simulated. Key-Words: - Bursting oscillations, chaotic, coupled neurons, fast-slow dynamics, Hindmarsh Rose system, synchronization. Introduction In the brain, sensory signals are transformed into various biophysical variables, such as membrane potential and firing rates, which are subsequently used in various processes. An important element in this signaling process is the single neuron. A single neuron can be represented as an electrical circuit, build of different compartments consisting of capacitors, conductances and leak voltages. Each neuron has a resting state with corresponding resting potential V rest, which value can vary from as high as -30mV to as low as -90mV depending on circumstances. In this state the neuron is in a dynamical equilibrium. Ionic currents, particularly sodium and potassium current, are flowing across the membrane in such a way that the net current is zero. Applying some stimuli will force the neuron from this equilibrium and make the neuron excitable. If the stimuli is large enough such that a threshold value, the threshold potential V thres, is crossed, the neuron will generate action potentials and starts to fire. On the other hand, if the stimuli is such that V thres is not passed, the neuron will return to its resting state []. Physiological models of neurons such as Hodgkin-Huxley [] and Huguenard-McCormick models [3] were developed directly from experimental measurements, and the variables and parameters reflect physiological properties. However, in order to gain a deeper understanding of the principles underlying the neuronal dynamics by mathematical analysis, simplified models are more accessible. The well-known FitzHugh-Nagumo equations [4, 5] constitute a polynomial model of tonically firing neurons, derived from the Hodgkin- Huxley equations. Using a similar approach, Hindmarsh and Rose [6] developed a polynomial model of a bursting neuron from a thalamocortical neuron model with detailed ionic currents. Hindmarsh-Rose model The polynomial form Hindmarsh-Rose (HR) model is nonlinear system which has the form dx = + + dy dz 3 y ax bx z I t () = c dx y = ε (( sx g) z) where x is a voltage-like variable, y controls the voltage recovery after an action potential, and z describes the slow dynamics of an adapting current and a, b, c, d, g, ε, s are parameters and I is input. Usually the parameter I, which means the current ISBN:

2 that enters the neuron, is taken as a control parameter. The topological structure according eq. is shown in Fig.. depicted in Fig.. b) which are called bursts. If the number of spikes per burst is irregular; as shown in Fig.. c), it is referred to as chaotic bursting [7]. I() t 3 ax bx z y y x y x c y dx g a) b) z z ε z s Fig.. The topological structure of the HR system described by eq.. Caused by ionic currents flowing across the neuron s membrane inside the cell body, the membrane potential of the neuron will change over time resulting in electrical signals that propagate through the axon. These electrical signals can be classified in the following three different states: a) Resting: In absence of stimuli, e.g. no reception of neurotransmitters, the net ionic current flowing across the neuron s membrane is zero and the membrane potential is constant. All neurons have negative resting potential, typically between 90 mv and 30 mv. b) Spiking: If a neuron receives neurotransmitters, i.e. it is being stimulated, its membrane potential will change. First, given the stimulus is of the excitatory type, due to excitatory ionic currents the membrane potential becomes more positive. At a certain point inhibitory ionic currents will dominate the excitatory currents and the membrane potential starts to decrease. The result is an action potential or spike. When a neuron produces successive spikes it is referred to as tonic spiking, shown in Fig.. a). c) Bursting: Instead of tonic spiking a neuron can also produce successive spike trains followed by some relatively long period of quiescence (as c) Fig.. The different dynamic states of the neuron. Top input signal I(t), bottom x(t) response. a) Spiking b) Bursting c) chaotic bursting Fig.3. The 3-dimensional projection of the HR system described by eq.. Example of the 3-dimensional projection of the HR system (according eq. ) and the time evolution is shown in Fig. 3 and Fig. 4, for parameters: a=, b=3, c=, d=5, g=-.6, ε=0.005, s=4, I(t)=.9 and initial conditions: [ ]. ISBN:

3 y = +exp - (x ( λ ϕ) ) (5) Fig. 4. The time evolution of the HR system; described by eq.. 3 Bidirectional Synchronization Interacting bursting neurons can exhibit basically two types of common rhythmic bursting: synchronization of bursts, where the neurons burst at the same time, regardless of the further evolution of their spikes; and complete synchronization [9], which involves also synchronization of spikes. In this part, the complete bidirectional synchronization of N neurons is described and simulation for 3 neurons is presented [0]. The network composed by N HR neurons, coupled by variable x, can be modeled by the system of equations [] as dxi dy dzi 3 = yi axi + bxi zi + I() t f( xi, xj) c dx y i = i i = ε (( sxi g) zi) () Fig. 5. Y =f(x) according eq. 5, for λ=[, 4, 6, 8] and φ=0.5. Fig. 6. The 3-dimensional projection of the HR systems left: x, y, z, right: x 3, y 3, z 3. Fig. 7. Example of x according x for k S =0.7 where i j, i=, N, j= N and coupling function f(x i,x j ) is given by f ( x, x ) = ( x V ) k Q( x ) (3) i j i R S j where V R is reversal potential, k S is coupling strength and Q(x j ) is N Qx ( j) = w (4) +exp - (x ( λ ϕ )) ij j= j S where w ij is connectivity (w ij = - connected or w ij = 0 - not connected), λ is characteristic of neuron cell and φ S is threshold of action potential for a neutron. Graph of the coupling function according eq. (5) is shown in Fig. 5. Fig. 8. Time evolution of state variable x, x and x 3. The simulation results for 3 neurons (bidirectional synchronized) according eq., 3 and 4 are shown in Fig. 6, 7, 8 and 9, for parameters: ISBN:

4 a=, b=3, c=, d=5, g=-.6, ε=0.004, s=4, I(t)=.9, λ=6.0, φ=-0.5 and different initial conditions. From Fig. 9 can be seen that all 3 chaotic systems are synchronized for time>400. the value of parameter a, each cell demonstrates two qualitatively different regimes of chaotic behavior: chaotic bursts (Fig. 0 a), b), c)) and continuous chaotic oscillation; see Fig. 0 d). Such dynamics resemble the bursting dynamics measured in the experiments with biological neurons. a /( + ( ) ) ε / N x n + z Fig. 9. The time evaluation of synchronization errors between 3 neurons: x x and x 3. y n z y n + c b () (3) ( N) synchronization inp. Fig.. Block diagram of the one neuron with synchronization inputs and output. Fig. 0. The waveforms ( ) of the chaotic behavior of single (discrete time) neuron cell, computed for different values of a, with c = b = a) a=4., b) a=4., c) a=4.4, d) a= Bursting Discrete Time System In this part the simplification of continuous HR model is obtained by using two-dimensional discrete-time systems, like the model proposed by [] x a = + y + xn y = y cx b n n n (6) where is the fast and y n is the slow dynamical variable. The slow evolution of y n is due to the small values of the positive parameters b and c, which are on the order of The value of parameter a is selected in the region a>4.0, where the map produces chaotic oscillations in. Depending on 5 Discrete Systems Synchronization In this example, burst synchronization which is weaker than complete synchronization and thus easier to achieve is simulated [3]. The existence of a slow time scale in coupled bursting neurons enables to define a bursting phase and frequency, even though on the spiking time scale they behave asynchronously. The existence of coherent bursting may be regarded as an example of chaotic phase synchronization, which is a widely investigated phenomenon in a variety of physical and biological systems [4]. Chaotic phase synchronization is defined as the occurrence of a certain relation between phases of interacting systems, bursting neurons in our case, while the amplitudes (related to the spiking time scales) can remain chaotic and uncorrelated [5]. The i-th discrete time system is described by eq. (7) a ε x y x + N ( j) = + n + n N j= ( xn ) y = y cx b n n (7) ISBN:

5 where x is the fast and y is slow dynamical n variables, ε is the strength of coupling and N is number of coupled neurons. The other parameters a (i), b and c are the same as for (6). The block diagram of one neuron with synchronization inputs and output is shown in Fig.. The simulation results for two neurons described by eq. (7) are shown in Fig., 3 and 4. Fig.. Time evolution of slow variables Fig. 3. Time evolution of fast variables. n displayed. In Fig. 4, the synchronization error of slow variable is shown. 6 External Synchronization It is important to note, that neurons can be synchronized also by external periodic signal. Such kind of external stimulation of brain has been extensively studied with respect to potential application to the control of pathological rhythms, since the synchronization of individual neurons is thought to play a key role in Parkinson s disease, essential tremor, and epilepsies [6]. Brain electromagnetic activity is a feature of neuronal network function in various brain regions. Modern neurophysiological techniques, such as electroencephalography (EEG) and event-related potentials (ERPs), are useful tools in the investigation of brain cognitive function in normal and pathological aging with an excellent time resolution. These techniques can indeormal and abnormal brain aging analysis of corticocortical connectivity and neuronal synchronization of rhythmic oscillations at various frequencies. For external synchronization the eq. (7) is extended by N a ε ( j) x = + y ( ) n + xn + Asin ωn + N j= (8) ( xn ) y = y cx b n n The simulations concerning external synchronizations will published in next paper. 7 Conclusion In this paper, continuous and discrete-time version of bursting oscillation neurons were simulated. The synchronization of 3 neurons (continuous) and neurons (discrete-time) bidirectional coupled with gap junction was modeled and simulated. It was shown, that exists of frequency locking between the bursting and driving phases. The wih of the frequency-locking interval increases with the driving amplitude up to a saturation due to chaotic evolution associated with the bursting dynamics. Fig. 4. Time evolution of synchronization error between slow variables. In Fig. and 3, the time evolutions of slow and fast variables of synchronized neurons are Acknowledgment This research was supported by the European Regional Development Fund and Ministry of Education, Youth and Sports of the Czech Republic under project No. CZ..05/..00/ : Regional Innovation Centre for Electrical Engineering (RICE) ISBN:

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