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Theoretical Neuroscience

Theoretical Neuroscience. Physics 405, Copenhagen University Block 4, Spring 2007 John Hertz (Nordita) Office: rm Kc10, NBI Blegdamsvej Tel 3532 5236 (office) 2720 4184 (mobil) hertz@nordita.dk www.nordita.dk/~hertz/course.html Texts:

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Theoretical Neuroscience

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  1. Theoretical Neuroscience Physics 405, Copenhagen University Block 4, Spring 2007 John Hertz (Nordita) Office: rm Kc10, NBI Blegdamsvej Tel 3532 5236 (office) 2720 4184 (mobil) hertz@nordita.dk www.nordita.dk/~hertz/course.html Texts: P Dayan and L F Abbott, Theoretical Neuroscience (MIT Press) W Gerstner and W Kistler, Spiking Neuron Models (Cambridge U Press)http://diwww.epfl.ch/~gerstner/SPNM/SPNM.html

  2. Outline • Introduction: biological background, spike trains • Biophysics of neurons: ion channels, spike generation • Synapses: kinetics, medium- and long-term synaptic modification • Mathematical analysis using simplified models • Network models: • noisy cortical networks • primary visual cortex • associative memory • oscillations in olfactory circuits

  3. Lecture I: Introduction ca 1011 neurons/human brain 104/mm3 soma 10-50 mm axon length ~ 4 cm total axon length/mm3 ~ 400 m

  4. Cell membrane, ion channels, action potentials Na in: V rises, more channels open  “spike” Membrane potential: rest at ca -70 mv Na-K pump maintains excess K inside, Na outside

  5. Communication: synapses Integrating synaptic input:

  6. Brain anatomy: functional regions

  7. Visual system General anatomy Retina

  8. Neural coding: firing rates depend on stimulus Visual cortical neuron: variation with orientation of stimulus

  9. Neural coding: firing rates depend on stimulus Visual cortical neuron: variation with orientation of stimulus Motor cortical neuron: variation with direction of movement

  10. Neuronal firing is noisy Motion-sensitive neuron in visual area MT: spike trains evoked by multiple presentations of moving random-dot patterns

  11. Neuronal firing is noisy Motion-sensitive neuron in visual area MT: spike trains evoked by multiple presentations of moving random-dot patterns Intracellular recordings of membrane potential: Isolated neurons fire regularly; neurons in vivo do not:

  12. Quantifying the response of sensory neurons spike-triggered average stimulus (“reverse correlation”)

  13. Examples of reverse correlation Motion-sensitive neuron in blowfly Visual system: s(t) = velocity of moving pattern in visual field Electric sensory neuron in electric fish: s(t) = electric field Note: non-additive effect for spikes very close in time (Dt < 5 ms)

  14. Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval

  15. Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t)

  16. Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t)

  17. Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t) Probability of firing for the first time in [t, t + Dt)/ Dt :

  18. Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t) Probability of firing for the first time in [t, t + Dt)/ Dt :

  19. Spike trains: Poisson process model Homogeneous Poisson process: = rate = prob of firing per unit time, i.e., prob of spike in interval Survivor function: probability of not firing in [0,t): S(t) Probability of firing for the first time in [t, t + Dt)/ Dt : (interspike interval distribution)

  20. Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T):

  21. Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T): Probabilityof exactly 2 spikes in [0,T):

  22. Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T): Probabilityof exactly 2 spikes in [0,T): … Probabilityof exactly n spikes in [0,T):

  23. Homogeneous Poisson process (2) Probabilityof exactly 1 spike in [0,T): Probabilityof exactly 2 spikes in [0,T): … Probabilityof exactly n spikes in [0,T): Poisson distribution

  24. Poisson distribution Pprobability of n spikes in interval of duration T:

  25. Poisson distribution Pprobability of n spikes in interval of duration T: Mean count:

  26. Poisson distribution Pprobability of n spikes in interval of duration T: Mean count: variance:

  27. Poisson distribution Pprobability of n spikes in interval of duration T: Mean count: variance: i.e., spikes

  28. Poisson distribution Pprobability of n spikes in interval of duration T: Mean count: variance: i.e., spikes large :  Gaussian

  29. Poisson distribution Pprobability of n spikes in interval of duration T: Mean count: variance: i.e., spikes large :  Gaussian

  30. Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution:

  31. Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution: Mean ISI:

  32. Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution: Mean ISI: variance:

  33. Poisson process (2): interspike interval distribution (like radioactive Decay) Exponential distribution: Mean ISI: variance: Coefficient of variation:

  34. Poisson process (3): correlation function Spike train:

  35. Poisson process (3): correlation function Spike train: mean:

  36. Poisson process (3): correlation function Spike train: mean: Correlation function:

  37. Stationary renewal process Defined by ISI distribution P(t)

  38. Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t):

  39. Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t): define

  40. Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t): define

  41. Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t): define Laplace transform:

  42. Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t): define Laplace transform: Solve:

  43. Fano factor spike count variance / mean spike count for stationary Poisson process

  44. Fano factor spike count variance / mean spike count for stationary Poisson process

  45. Fano factor spike count variance / mean spike count for stationary Poisson process 

  46. Fano factor spike count variance / mean spike count for stationary Poisson process  for stationary renewal process (prove this)

  47. Nonstationary point processes

  48. Nonstationary point processes Nonstationary Poisson process: time-dependent rate r(t) Still have Poisson count distribution, F=1

  49. Nonstationary point processes Nonstationary Poisson process: time-dependent rate r(t) Still have Poisson count distribution, F=1 Nonstationary renewal process: time-dependent ISI distribution = ISI probability starting at t0

  50. Experimental results (1) Correlation functions

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