1. Neural chaos (Chapter 11 of Wilson 1999) Klaas Enno Stephan Laboratory for Social & Neural Systems Research Dept. of Economics University of Zurich Wellcome Trust Centre for Neuroimaging Institute of Neurology University College London Computational Neuroeconomics and Neuroscience University of Zurich, 4 May 2011

2. Recap: Oscillations & limit cycles Oscillation: A trajectory X(t) of a dynamical system with any number of dimensions is an oscillation if: X(T+t) = X(t) for some unique T > 0 and all t. Limit cycle: An oscillatory trajectory in the state space of a nonlinear system is a limit cycle if all trajectories in a sufficiently small region enclosing the trajectory are spirals. If these neighboring trajectories spiral towards the limit cycle as t , then the limit cycle is asymptotically stable. If neighboring trajectories spiral away from the limit cycle as t , the limit cycle is unstable. Stable limit-cycles imply self sustained oscillations. Any (small) perturbation from the closed trajectory would cause the system to return to the limit cycle, making the system stay on the limit cycle. stable limit cycle unstable limit cycle

3. Recap: Poincaré-Bendixon Theorem PB Theorem: If an annulus can be constructed in a two-dimensional system such that all trajectories enter it, yet it contains no steady states, then a limit cycle must exist within the annulus. Why not simply extend this to ≥3 dimensions by replacing the annulus with a torus? In a deterministic autonomous system, trajectories can never cross, and this provides a definitive constraint in two dimensions. However, trajectories in a system with ≥2 dimensions can pass by without intersecting in an infinite number of ways. –quasiperiodic trajectories –aperiodic (chaotic) trajectories

4. Definition: quasiperiodic trajectories system with ≥3 dimensions that has at least two frequencies, at least one being irrational the trajectory of such a system never rejoins or intersects itself however, quasiperiodic trajectories are similar to periodic trajectories and two quasiperiodic trajectories starting from similar initial conditions will remain close to each other demo: quasiperiodic.m

5. Definition of chaos A deterministic nonlinear dynamical system in three or more dimensions exhibits chaos if all of the following conditions are satisfied for some range of parameters: 1.Trajectories are aperiodic (not quasiperiodic). 2.These aperiodic trajectories remain within a bounded volume of the state space but do not approach any steady states. 3.There is sensitivity to initial conditions such that arbitrarily small differences in initial conditions between nearby trajectories grow exponentially in time. NB: this refers to systems which are deterministic, nonlinear, and with ≥ 3 dimensions

6. Chaos =deterministic behaviour that is very sensitive to initial conditions NB: chaotic systems are deterministic systems! they are perfectly predictable given knowledge of the initial conditions however, in real life, we never know the initial conditions precisely; even tiny errors in measuring initial conditions will produce trajectories which wildly differ example: weather forecasting

7. Example of a chaotic system: The Lorenz oscillator

8. The Lorenz oscillator x y z strange attractor = the orbit is bounded, but neither periodic nor quasiperiodic and not convergent

9. Demo: dependence on initial conditions red and blue trajectories: change in the initial conditions by 10 -5 demo: Lorenz_mod_2.m

10. Testing for the presence of chaos Qualitative criteria that can be applied to measured time series –Fourier power spectrum –first return map (Poincaré map) Quantitative criterion that requires knowing the system’s differential equation –Lyapunov exponent

11. Fourier transform of non-chaotic systems random periodic quasiperiodic

12. Fourier transform of the Lorenz oscillator qualitative signature of chaos: very complex spectrum with all frequencies present and with power decreasing with frequency

13. Poincaré map intersection of a periodic orbit with a lower dimensional subspace (Poincaré section) that is transversal to the flow of the system at a point p can be interpreted as discrete dynamical system with a state space that is one dimension smaller than the original continuous dynamical system preserves many properties of the original system and is thus sometimes used for analyzing the original system conceptual basis for computing what Wilson refers to as a “first return map” in one dimension

14. First return map of the Lorenz oscillator First return times wrt. to z 0 : Descriptively: one chooses a value z 0 and measures all the intervals between the times when z(t) returns to this same value First return map: A two-dimensional graph which plots   i+1 as a function of  i For pure oscillations, the first return map consists of a single point (equal to the period). First return map of the Lorenz oscillator for z 0 =25

15. Lyapunov exponent (LE) expresses the dependency on initial conditions: average exponent of rate of divergence e  t of neighboring trajectories requires knowing the system’s equations one LE for each dimension of the system; due to exponential growth, only the largest one needs to be considered if largest LE is positive, then the system is chaotic

16. Chaotic behaviour of Hodgkin-Huxley neurons parameter A controls amplitude of injected sinusoidal current implicitly a four-dimensional system –sine term is the solution of two linear 1 st order ODEs V R

17. Chaotic behaviour of HH neurons chaotic behaviour for A=0.007 (and I 0 =0.075 and  =264.6 Hz) Lyapunov exponent = 0.16 this will produce divergence by a factor of e 16 =  10 7 over 100 ms V 0 = -0.7 V 0 = 0

18. Implications for modelling neuronal systems if a neural system is in a chaotic regime, we cannot predict details of spike trains beyond very brief intervals (determined by the Lyapunov exponent) this cannot be overcome by including ever more biophysical details in the model! only two options: –be content with predicting only the immediate future –predict coarser states (e.g. mean firing rate over time periods or over neurons) cf. weather prediction: detailed predictions impossible beyond a few days, but monthly averages can be predicted accurately far into the future

19. Implications for modelling neuronal systems “Neuroscientists who believe that it is necessary to understand in detail all of the ionic currents of every neuron in a network to make predictions should recognize that greater descriptive detail in a chaotic neural system will not generally lead to greater predictive power.” H.R. Wilson, p. 183

20. Some philosophical thoughts On a more philosophic level, what does neural chaos have to say about our own brains and thought processes?... First, chaos may provide a limitation on each individual's ability to predict his/her own behavior in detail... Although we know very little about the neural processes involved in thinking or consciousness, we do know that they involve neurons that might sometimes operate in chaotic regimes. Even though the brain may be totally deterministic, therefore, we may sometimes have no idea why we suddenly perform an unexpected act or make a snap decision! Thus, the old free-will versus determinism controversy in philosophy may have its resolution in neural chaos: a totally deterministic brain may nevertheless produce behaviors that are not predictable either by that brain itself or by any other brain on the planet!” H.R. Wilson, p. 184

21. Thank you