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STOCHASTIC MODELS IN NEUROSCIENCE

STOCHASTIC MODELS IN NEUROSCIENCE. MARSEILLE, FRANCE JANUARY 2010. THANKS TO MICHELE, NILS AND SIMONA FOR ORGANIZING SUCH AN INTERESTING AND ENJOYABLE MEETING. THE EFFECTS OF WEAK NOISE ON RHYTHMIC NEURONAL ACTIVITY. HENRY C. TUCKWELL

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STOCHASTIC MODELS IN NEUROSCIENCE

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  1. STOCHASTIC MODELS IN NEUROSCIENCE MARSEILLE, FRANCE JANUARY 2010

  2. THANKS TO MICHELE, NILS AND SIMONA FOR ORGANIZING SUCH AN INTERESTING AND ENJOYABLE MEETING.

  3. THE EFFECTS OF WEAK NOISE ON RHYTHMIC NEURONAL ACTIVITY HENRY C. TUCKWELL MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES, INSELSTRASSE 22-26, LEIPZIG, GERMANY

  4. PLAN OF TALK WEAK NOISE EFFECTS ARE CONSIDERED IN SINGLE NEURON MODELS • THE HODGKIN-HUXLEY SYSTEM OF ODE’S (POINT MODEL) • HH AS PACEMAKER (ODE) C. A MULTI-COMPONENT BIOLOGICAL PACEMAKER MODEL D. THE HODGKIN-HUXLEY SYSTEM OF PDE’S (CABLE MODEL)

  5. A: HH ODE MODEL WITH NOISEBACKGROUND • We had been considering the effects of noise on coupled type 1 (QIF) neurons ( See Gutkin, Jost & Tuckwell, Theory in Biosciences 127, 135-139 (2008) & Europhysics Letters 81, 20005 (2008) ) and commenced a similar study of coupled HH neurons. However, finding that minima arose in the activity, at various values of the coupling strength, and especially zero coupling, led us to consider single HH neurons with noise.

  6. Introduction • Recall that noise may induce firing. e.g. a leaky integrate and fire model (but not an integrate and fire model) neuron with periodic impulsive excitation at frequency f with amplitude a>0 may never fire an action potential, but if the input is Poisson with the same mean frequency f>0 , the neuron will fire in a finite time with probability 1 – see Introduction to Theoret Neurobiol Ch 3 and later chapters. • Also, noise even of zero mean increases the mean membrane potential in the standard LIF model. • However, here we are mainly concerned with the inhibitory effects of noise on neuronal spiking.

  7. For the HH ODE model with additive noise

  8. USING STANDARD PARAMETER VALUES THE CRITICAL VALUE OF µ TO INDUCE REPETITIVE FIRING (HOPF BIFURCATION) IS ABOUT 6.44. FOR VARIOUS VALUES OF µ AND σ ONE OBTAINS RESULTS SUCH AS THESE (µ=6.6)

  9. WITH CORRESPONDING ORBITS

  10. WE INVESTIGATED THE DEPENDENCE OF THE NUMBER OF SPIKES OVER A LIMITED TIME PERIOD ON THE NOISE AMPLITUDE FOR VARIOUS VALUES OF THE MEAN CURRENT AND OBTAINED THE FOLLOWING IN THE ADDITIVE NOISE CASE.

  11. HH, ADDITIVE WHITE NOISESPIKES OVER 1000 MSEC

  12. FOR THE CONDUCTANCE-BASED INPUT WE HAVEAN INPUT CURRENT OF THE FORMI_c=g_E(V_E-V) + g_I(V_I-V)where V_E, V_I are the excitatory and inhibitory reversal potentials

  13. with the following results:

  14. WE ALSO CONSIDERED SWITCHING THE NOISE ON AT A RANDOM TIME AFTER REPETITIVE SPIKING WAS ESTABLISHED

  15. In addition we consideredchoosinginitial conditions randomly with result as shown here

  16. THE APPROXIMATE BASINS OF ATTRACTION OF THE REST POINT AND LIMIT CYCLE (WHICH DEPEND ON MU)

  17. CONSIDER HOW THE PROCESS LEAVES THE LIMIT CYCLE

  18. WE COLLECTED DATA ON WHERE THE EXIT POINTS WERE ROUGHLY LOCATED, WHICH CAN BE COMPARED WITH THE ESTIMATE OF THE BASINS OF ATTRACTION

  19. WHEN THE NOISE IS “LARGE” THE PROCESS MAY QUITE FREQUENTLY MAKE TRANSITIONS TO AND FROM THE BASINS OF ATTRACTION OF THE STABLE POINT AND THE LIMIT CYCLE

  20. WITH INTERSPIKE INTERVAL HISTOGRAMS AS FOLLOWS

  21. One sees that there is a “competition” between the tendency of noise to stop the spiking and the tendency for it to induce spiking.

  22. MARKOV THEORY WE CONSIDER THE NATURE OF THE ATTRACTORS OF WHICH, FOR MEAN CURRENTS NOT MUCH GREATER THAN THE CRITICAL VALUE, THERE ARE TWO : A STABLE REST STATE AND A STABLE LIMIT CYCLE. JUST PAST THE CRITICAL VALUE THE BASIN OF ATTRACTION (BOA) OF THE LIMIT CYCLE IS SMALL AND A SMALL NOISY SIGNAL (OR ANY) CAN KICK THE DYNAMICS INTO THE BOA OF THE STABLE REST POINT – THUS TERMINATING THE SPIKING.

  23. Theory: Exit-time theory for Markov processes • Theorem: The process switches from spiking to non-spiking states (and vice-versa) in a finite time with probability one. The expected times which the system remains in one or the other state are the solutions of linear partial differential equations given below • Sketch proof • The process (V,m,h,n) has an infinitesimal operator L. That is, the transition density p satisfies a Kolmogorov equation • ∂p/ ∂ t = Lp

  24. The prob pL of leaving the BOA BL of the limit cycle satisfies • LpL =0 on BL (*) • with boundary condition • pL =1. • The solution of * is pL = a constant. Hence, because process is continuous, pL =1 throughout BL. • Similarly for the prob pR of leaving the BOA BR of the rest state. Standard theory gives that the expected time to stay in the spiking state satisfies LFL =-1 on BL with boundary condition FL = 0. Similarly for the expected time to leave BR. The behaviour of the system is thus characterized by a sequence of alternate exit times from BL and BR.

  25. MOMENT ANALYSISWe have also sought explanations of these phenomena “analytically” . Thus we have found the moment equations for an HH neuron with noise – in the additive noise case there are 14 de’s.

  26. FOR HH THE COMPONENTS ARE, USING STANDARD SYMBOLS, X1=V, X2=n, X3=m AND X4=h. • The means are denoted by m_1, m_2, m_3, m_4 and there are 4 variances C_11, C_22, C_33 and C_44 together with another 6 covariances C_12, C_13, C_14, C_23, C_24, C_34 = 14 1st and 2nd order moments. For example, C_24 = Cov (n(t), h(t))

  27. FAIRLY LENGTHY CALCULATIONS GIVE FOR EXAMPLE, THE DE’S FOR THE MEAN AND VARIANCE OF THE ,VOLTAGE VARIABLE

  28. OTHER EQUATIONS INVOLVE THE ALPHA’S AND BETA’S AND THEIR DERIVS E.G.

  29. FOLLOWING SHOW THE MEAN AND VARIANCE OF THE VOLTAGE: THE FIRST TWO SETS OF RESULTS ARE FOR SMALL NOISE AND SHOW THE EXCELLENT AGREEMENT BETWEEN ANALYTICAL AND SIMULATION RESULTS

  30. THIS SHOWS HOW A SMALL NOISE MAY SOON GIVE RISE TO A LARGE VARIANCE AND POSSIBLY DRIVE THE SYSTEM TO REST

  31. THIS IS FURTHER ILLUSTRATED BY THIS SAMPLE PATH PICTURE. WE HAVE ALSO DETERMINED THAT THE SPEED OF THE PROCESS IS SMALL NEAR THE EXIT POINT. THE REDUCTION IN VELOCITY CONTRIBUTES TO THE EXIT.

  32. EXPERIMENTAL CONFIRMATION OF THE SILENCING OF NEURONAL ACTIVITY BY NOISE CAME IN 2006 ON SQUID AXON – AN ARTICLE BY Paydarfar, Forger & Clay: Noisy inputs and the induction of on-off switching behavior in a neuronal pacemaker. J. Neurophysiol. 96, 3338-3348. 8 AXONS WERE EXAMINED.

  33. B. HH AS “PACEMAKER” Writing the HH auxiliary equations in the following form with activation and inactivation steady state values and time constants as a function of voltage, one may turn the HH neuron into a spontaneously firing “cell” by shifting, for example, the half activation potential to -30.5 mV from about -28.4 mV (assumed resting at -55 mV).

  34. Normally the HH neuron does not fire with zero input or with hyperpolarizing input current, but with V_1/2 = -30.5 there is a threshold for repetitive spiking around +1.8 nA (standard model). See below.

  35. Now with an initial value V(0)=V_R + 10, periodic spike trains were impeded with noise with a similar behaviour as in the standard HH case.

  36. C. BIOLOGICAL PACEMAKER WITH NOISE • BACKGROUND • Interest here focuses on whether pacemaker activity, which often performs vital biological functions, could be stopped by a small noise. • I had been studying the brain circuitry involved in stress, which involves interactions between the nervous and endocrine systems. • The structures involved are many and diverse and only a partial picture is able to be given. • I have put together as a first attempt a map of the brain components involved.

  37. MODELS • Many pacemaker or pacemaker-like models have been proposed over the last 18 years including • McCormick and Huguenard 1992 Thalamic relay • Schild et al. 1993 Nucleus tractatus solitarii (brainstem) • Destexhe et al. 1994 Thalamic reticular nucleus • Rybak et al. 1997 Respiratory neurons…….. • up to more recently….. • Rhodes and Llinas 2005 Thalamic relay • Putzier et al 2009 Dopamine neurons in SN (empirical with virtual • L-type channels) • See also “Thalamocortical Assemblies”, Destexhe & Sejnowski 2002

  38. The are 10 currents included in the present model. Except for I_K(Ca), these channels are all voltage-dependent in the present model. • In addition the internal calcium concentration Ca_i is included in the model. • This is determined by some calcium currents with allowances for buffering and pumping. • The activation and inactivation curves are taken from various references. The 10 maximal conductances are estimated from area measurements and channel densities. In total about 75 parameters must be specified.

  39. Model with noise+ steady current (which could be synaptic in origin) • CdV/dt = -Σ_i g_i (V-V_i) + -mu + σw where mu is a constant, g_i is conductance, w is GWN and the sum is over the above 10 channel-types. V is membrane potential, not depolarization.

  40. Rhythmic firing near threshold- 10-component process

  41. Ten trials small noise

  42. Ten trials larger noise

  43. Ten trials with sigma = 0.6

  44. D: THE HODGKIN-HUXLEY PDES WITH NOISE

  45. RESULTS 1 : no noise: X_1=0.2, L=6 VARIOUS MU. CURRENT LEFT ON.

  46. RESULTS 2: NUMBER OF SPIKES ON (0,L) UP TO T=160 FOR TWO VALUES OF X_1: BIFURCATION TO REPETITIVE FIRING AT ABOUT MU=6

  47. Results 3: Inhibitory effects of noise applied throughout whole cylinder: Signal to x_1=0.1. E[N]=expected number of spikes on (0,L) at t=160 ms. 50 trials..

  48. RESULTS 4: A TRIAL WITH LARGER NOISE SHOWING HOW SECONDARY SPIKES MAY LEAD TO AN END RESULT WITH NO SPIKE ON (0,L)

  49. RESULTS 5 :NOISE ON SMALL INTERVALS

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