Neural Excitability in a Subcritical Hopf Oscillator with a Nonlinear Feedback Gautam C Sethia and Abhijit Sen Institute for Plasma Research, Bhat, Gandhinagar 382 428, INDIA
Motivation Neural Excitability which is the property responsible for generation of action potentials in neurons is an active area of research. Our goal is to study the effect of time delayed feedback on a neural system.
What is Neural Excitability? A neuron is said to be excitable if a small perturbation away from a stable equilibrium state can result in a large excursion of its potential before returning to its original state. This phenomenon is known as spiking of neurons. If the system s activity alternates between a rest state and a state of repetitive spiking, the system is said to exhibit bursting behavior.
Bifurcations for Neural Excitability: These large excursion exist because the system is close to bifurcations from rest to oscillatory states. The two bifurcations responsible for this transition are: Saddle-Node bifurcation on a limit cycle and Hopf Bifurcation.
Bifurcations giving rise to Neural Excitability: Izhikevich Eugene M., Neural Excitability, Spiking and Bursting, Int. J. Bif. Chaos Vol. 10, No.6 (2000) 1171-1266
Bifurcations in Hodgkin-Huxley Model: The Hodgkin-Huxley model, as well as many other biophysical neuron models, has a typical bifurcation structure where, as the bifurcation parameter increases, stable and unstable limit cycles appear via fold limit cycle bifurcation. The latter shrinks down to the rest state and makes it loose its stability via subcritical Hopf bifurcation. A topological normal form of a subcritical Hopf bifurcation oscillator qualitatively illustrates the above features.
The Hodgkin-Huxley Model: Typical Bifurcation Structure Izhikevich Eugene M., Neural Excitability, Spiking and Bursting, Int. J. Bif. Chaos Vol. 10, No.6 (2000) 1171-1266
What do we study? We numerically study the dynamical properties of a normal form of subcritical Hopf oscillator (at the Hopf bifurcation point) subjected to a nonlinear time delayed feedback. We choose the non-linearity to be quadratic in nature. We further investigate the spiking/bursting properties of our model.
Why feedback and delay? We have introduced a quadratic nonlinear feedback which is the simplest and the lowest order nonlinearity that provides an excitable behavior in our model. The feedback is time delayed to account for finite propagation times of signals. The self-feedback term can mock up a variety of physical effects. In a collection of neurons, the term can be regarded as a source term representing the collective feedback due to the rest of the neurons.
AUTAPSE Neurons: A more direct and natural application could be in the modeling of autapse neurons (neurons with auto-synapses) where the feedback term can represent the phenomenon of signals looping back on the neuron through axons that close in on the neuron s own dendrites.
Normal form of a Bautin bifurcation oscillator is: z& ( t) where a b is β is = ( a the Hopf z the + i( ω + β z( t) = x + bifurcation parameter and bifurcation shear iy is parameter complex variable determinesthe supercritical( b < ( β determinesthe dependence of a 2 ) + b z( t) 2 z( t) 4 ) z( t) 0) or subcritical (b > 0) frequency on amplitude)
The Hopf Oscillator with time delayed nonlinear feedback: z& ( t) where k andτ is β = a = = ( i( ω + β z( t) -0.5for all simulations 0 is and the oscillator without the feedback is poised at subcritical the time b = 1 Hopf 2 ) + feedback delay z( t) so that the strength z( t) bifurcation point. 2 4 ) z( t) 2 kz (t τ)
Bifurcation diagrams as a function of the feedback strength k :
Two parameter bifurcation diagram in feedback strength (k) and the time delay τ :
Different type of possible bursting in a neural system near saddle-node separatrix loop bifurcation : (from Izhikevich 2000)
Different type of possible bursting in a neural system near Bautin bifurcation : (from Izhikevich 2000)
Saddle-Node on a Limit Cycle (SNLC) Bifurcation : The feedback strengths k=0.4252 and 0.425 respectively.
Typical trajectories: solid:τ=0; dash-dot: τ=0.5; dash: τ=0.57
The temporal response of the model to an external stimulus: z& ( t) 2 2 = ( i( ω + β z( t) ) + z( t) z( t) ) z( t) kz ( t e i Ω + ε t + 2Dξ ( t) 2 4 τ ) Where ε is the amplitude and Ω is the frequency of the external signal and ξ(t) is the zero mean Gaussian white noise with intensity D. The time period T of the external signal is 2π/Ω. Both the signal as well as the noise are subthreshold and thus do not give rise to spiking on their own.
What are ISIH, ISI and IMS? The Inter-Spike Interval Histogram (ISIH); in which the time intervals between successive spikes are assembled into a histogram, is found to be useful in characterizing the spiking pattern of a neuron. ISIH typically exhibit multimodal structure with peaks at integer multiples of a basic Inter-Spike Interval (ISI); a feature generally referred to as Integer Multiple Spiking (IMS).
A segment of a typical time series exhibiting spiking. The continuous trace is for τ=0 and the dashed trace is for τ=0.5 K=0.45, ε=0.04, D=0.004, Ω=0.1 (T 62.8) and β=-0.5
The interspike interval histograms (ISIHs) with delay (dashed curve) and without delay (solid curve):
The rate of change of phase (dφ/dt) as a function of phase (φ) near the bifurcation point :
Peak profile of an individual spike in the absence/presence of time delay :
t i e t kz t z t z t z t z i t z Ω + + + = ε τ β ω ) ( ) ( ) ) ( ) ( ) ) ( ( ( ) ( 2 4 2 2 & Simulation of bursting by adding an external stimulus to the model: where ε (=0.02) is the amplitude and Ω (=0.01) is the frequency of the external signal. The time period T of the external signal is 2π/Ω.
SNLC/SNLC (parabolic) bursting in the absence of delay:
SNLC/SNLC (parabolic) bursting with finite time delay (τ=0.3) but less than the threshold value :
SN/SSL (square wave) bursting in the presence of a large enough delay (τ=0.5) to be in the bistable region:
Conclusions: We have investigated the effect of time delay on the excitability properties of a single neuron with the help of a mathematical model consisting of subcritical Hopf oscillator with a nonlinear time delayed feedback. We find that time delay can have significant influence on the spiking properties of the neuron, such as in enhancing the frequency of spikes, triggering of multi-spikes (bursty behavior) and altering the fine structure of individual spikes.
Conclusions (contd.): All of this can have interesting practical implications in real biological systems. Our model neuron could also provide a useful paradigm for gaining more insight into the behavior of autapse neurons which are presently receiving a great deal of theoretical and experimental attention.
Conclusions about Bursting: The dynamics of our model is capable of exhibiting different types of bursting. A decrease in the feedback strength increases the number of spikes in a burst and also the width of the bursts. An increase in time delay also enhances the spike rate within a burst as well as the width of the bursts. The two parameter bifurcation diagram in k &τ space broadly explains the origin and characteristics of the different types of bursting activities.
Acknowledgements : We have made extensive use of the software packages XPPAUT (Bard Ermentrout) and DDE-BIFTOOL (K. Engelborghs, T. Luzyanina, and G. Samaey) for our studies.