Structure of Brain at Small, Medium and Large Scales. Laval University 4 May 2006

Transcription

1 Structure of Brain at Small, Medium and Large Scales Helmut Kröger Laval University 4 May 2006

2 Collaboration: Dr. Alain Destexhe,, Neurosciences Intégratives et Computationnelles, CNRS, Paris Dr. Igor Timofeev,, Département D de physiologie, Université Laval Students: Reza Zomorrodi, Tanguy Pallaver, Post-doc: Gurken Melkonyan,, Ph.D.

3 Part I: Brain at scale of single neuron. How is information transmitted from one neuron to the other? The spiking neuron in the Hodgkin-Huxley Huxley model.

4 Giant axon of squid

5 Nerve cells in the brain are called neurons. There is an estimated number of 10^10 to 10^13 neurons in the human brain. Each neuron can make contact with up to several thousand other neurons. Neurones are the units to process information.

6 input Hillock The neuron computes: Input processed to Output

7 Schematic 3-dimensional cross section of a cell membrane.

8 Neural cell membrane with ion channels.

9 Sodium and potassium channels during spike creation

10 Recording from a neuron: membrane potential Actual activity of a neuron

11 How to build a model..

12 Eletrical circuit analogue

13 Active vs. passive membrane Outside Na K L Inside In active membrane, conductances may vary with time and as function of the membrane potential

14 Currents in an active membrane I = I + I + I + ext Na K L I C I ext Outside Ohm' s I Na = g Law : V - V Na ( V R V Na Na = ) Na = 1 g Na I Na R Na Na K L I I I K L C = g = g K L ( V ( V dv = C dt V V L K ) ) Inside dv C = g Na ( VNa V ) + g K ( VK V ) + g L ( VL V ) + dt I ext

15 Result: Hodgkin Result: Hodgkin-Huxley Model Huxley Model ext L L K K Na Na I V V g V V n g V V m h g dt dv C = ) ( ) ( ) ( 4 3 m m m m m m m m m m dt dm β α α β α τ τ + = + = + =, 1, h h h h h h h h h h dt dh β α α β α τ τ + = + = + =, 1, n n n n n n n n n n dt dn β α α β α τ τ + = + = + =, 1, ) ( 1 ) 0.01 (0.1 ) ( V n V n e V e V V = = β α ) ( 1 ) 0.1 (2.5 ) ( V m V m e V e V V = = β α 1 1 ) ( 0.07 ) ( = = V h V h e V e V β α

16 Action potential of cortical neuron: Experiment

17 Simulation from Hodgkin-Huxley Huxley model

18 Thalamocortex: Exp

19 Simulation

20 Action potential is basis of information exchange in brain and nervous system. Multiple sclerosis inflammatory desease of de-myelination of axon: malfunction of action potential transmission.

21 Degenerative disease: Multiple Sclerosis

22 Normal brain 1: brain hemispheric MR-T1 orbital g

23 Part II The Brain at medium scale: How are electrical signals transmitted in the cerebral tissue? Neural avalanches and 1/f scaling.

24 Neuron with synapse (forground( forground) ) + glial cells (attached)) + astrocyte (background)

25 Neurons (yellow)) + astrocyte (green)

26 A Region (3) Region (2) Region (4) Region (1) Region (5) B (1) (2) (3) (4) (5) Charge density 0 (center) Distance C D Source E Charge density 0 (center) Distance C. Bédard, B H. Kröger ger,, A. Destexhe,, (2004) Biophysics J., 86: C. Bédard, B H. Kröger ger,, A. Destexhe,, Phys. Rev.. E, in press,

27 Electric circuit model of extra- cellular medium T = RC R C T = RC C V ind V source Physical time scale: T Maxwell = ε σ

28 A B (A) Time-dependent source of period T. (B) Response of medium for different T_Maxw.

29 A Filtering of signal: (A) T_signal=10 ms, T_relax_filter=1ms. (B) T_relax_filter=10ms, (C) T_relax_filter=100ms. B C

30 A (s) A 100 (a) 50 B B C C Extracellular potential created by time-independent neuron (source) s, and glial (passiv) cell (a). (A) total electric potential, (B) source potential, (C) induced potential

31 A A B (a) (s) B C C D D Time-dependent neuron (source) s, and glial (passiv) cell (a). Frequency: 0,100,200,400 Hz in A,B,C,D, resp.

32 Neural Avalanches: Size Distribution Follows Power Law Follows Power Law The size distribution of neuronal avalanches in mature cortical cultured networks follows a power law with an exponent ~3/2 (dashed line). The data, re-plotted from Figure 4 of [30] shows the probability of observing an avalanche covering a given number of electrodes for three sets of grid sizes shown in the insets with n=15, 30 or 60 sensing electrodes (equally spaced at 200 μm). The statistics is taken from data collected from 7 cultures in recordings lasting a total of 70 hours and accumulating ( )avalanches per hour (mean +- SD)

33 Interpretation: Scaling behavior for waking synaptic current: f^0 (0-20 Hz) and 1/f^2 (20-65 Hz). Experiment: 1/f (0-20 Hz) and 1/f^3 (20-65 Hz). Conclusion: 1/f scaling due to matter of parietal tissue. Observation of 1/f scaling in nature: Resistors,, transistors, avalanches, infinite chain of coupled RC circuits Model of cortical tissue equivalent to RC circuit: C. Bédard, B H. Kröger ger,, A. Destexhe,, Phys. Rev.. E in press, physics/o Note: Temporal behavior of Inter-Spike Spike- Intervals: Poissonian. Inconsistent with model of Self-Organized Organized-Criticality.

34 Understanding electrical properties (frequency dependence attenuation, scaling) of cerebral tissue. Potential use: - diagnostic tools for early detection of degenerative brain diseases - important for constructing neuron- microchip interfaces.

35 Normal brain 1: brain hemispheric MR-T1 orbital g

36 Degenerative disease: Alzheimer 1

37 Degenerative disease: Creutzfeld-Jacob

38 Part III: The brain at large scale. Memory, learning and neural connectivity.

39 Small World Networks and Scale Free Networks The first papers: - D.J. Watts, S.H. Strogatz,, Nature 393(1998) R. Albert, H. Jeong,, A.L. Barbasi,, Nature 401 (1999) B.A. Huberman,, L.A. Adamic,, Nature 401 (1999) J.M. Kleinberg, Nature 406 (2000) S.H. Strogatz,, Nature 410 (2001) D.J. Watts, P.S. Dodds,, M.E.J. Newman, Science 296 (2002) 1302.

40 Milgram s Letter Experiment: Six Degrees of separation in organisation of human society. S. Milgram, The Small-World Problem, Psychology Today 1, (1967).

41 Short Average Path Length: L=7

42 High Local Clustering Coefficient C

43

44 What is a Small-World Network? Watts, D. J. and S. H. Strogatz. 1998, Nature 393: A Small-World network is a network architecture between a random network and a regular network. Starting from a regular network and at random rewiring some links to a far node yields SWN.

45 Why is high local clustering beneficial? Redundancy in case of breakdown of node. Small risk of error. Stability of system. Why is short path length beneficial? Rapid communication (Society: Milgram exp., WWW: fast search. Brain: fast response, coherent activation in motor cortex.)

46 Examples of small-world networks (1) Human society: - Milgram s letter experiment. - Organization of board members of big corporations. - Films actors network. - Paul Erdös (mathematician) publication network.

47 (2) Engineering: Grid of electrical power lines in western US. (3) Communication: - WWW, - Internet. (4) Biology: - Metabolic network of bacterium E.coli, - Neural network of nematode worm C. elegans.

48 Example of Network in Biology: Food Web of Fish in Ocean

49 Example of Scale-Free Network: Internet

50 Random vs. Scale-Free NW

51 SWN in Neuroscience: Experiments Cat cortex Macaque visual cortex. Human brain: Activity network from magnetic resonance imaging.

52 Scale-Free Brain Functional Netw. V.M. Eguiluz et al., cond-mat/

53 SWN in Neuroscience: Computational Models Fast response and coherence Efficient associative memory Fastest supervised learning Efficient self-organization of visual cortex (post-natal)

54 SWN of Hodgkin-Huxley Huxley Neurons Model: Hodgkin Huxley neurons in 1-D periodic network. Result: Fast response and coherent oscillations. L.F.Lago-Fernandez et al. Phys. Rev. Lett. 84 (2000) Possible relevance in neuroscience: Binding

55 SWN in Associative Memory Model Hopfield Model J.W. Bohland and A.A. Minai, Neurocomputing (2001) 489.

56 Hebb s learning rule. Patterns ζ of memory stored in synaptic weight w_ij. w ij = 1 N M μ = 1 ζ μ i ζ μ j Dynamical update rule of neuron S_i: McCulloch-Pitts network. S i ( t + 1) = sgn( w S ( t)) N j i ij j

57 Restoration of memory: Regular NN vs. SWN

58 Fastest Learning in SWN Model of visual cortex: Multi-layered layered feed-forward forward network (Perceptron( Perceptron). D. Simard, L. Nadeau, H. Kröger ger,, Phys. Lett.. A 336 (2005) 8.

59 -Regular : Each neuron is connected to all neurons in the next layer. -Random: Each neuron is connected randomly to a forward neuron, no backward connection is allowed. -Small-World: Starting from the regular architecture, some connections to the next layer are rewired to some forward layer

60 D_local and D_global vs. Connectivity Networks of 15 neurons per layers with 8 layers. Training with 100 patterns in 20 different runs. D_loc and D_glob are both small at 30 rewirings (few short-cuts): Small World architecture.

61 Minima of D_loc, D_glob and learning time coincide: SWN learns fastest.

62 Network of 5 Layers by 8 Neurons Simulation with 5 neurons per layers and 8 layers. NN was trained with 40 patterns for 50 different runs. Learning 40 patterns. The regular network almost fails to learn. With a few short-cuts the network learns well. The SWN architecture is better than regular and random random

63 Implications in Neurobiology Insight into complexity of organization of brain vs. functional tasks. Fastest learning guiding principle in evolution of nervous system of biological species? H. Kröger ger, Festschrift in Honour of M. El Naschie,, Springer, Wien (2005), p.98.

64 SWN and Self-Organization in Kohonen Network the Brain T. Pallaver,, H. Kröger ger,, M. Parizeau, National Project of Complex Data Structures, Fields Institute, Toronto 2005.

65 The Data Set

66 Evolution of neuron neighbor map

67 Data+ Representation by Neurons 1 Jeu

68 Dynamical cutting of connections Knowledge vs. error

69 Small World Network Regime

70 Results (1) Improved Kohonen Network gives reduced errors. (2) Knowledge function is good representation of actual knowledge/errors (3) Cutting connections is adapted to actual knowledge of system. (4) Cutting connections from beginning is biolgically realistic: In human brain there is very early onset of pruning the dendritic tree (also genetically controled death of neurons). (5) IMPORTANT OBSERVATION: Network has SWN architecture during most of the dynamic phase of self-organization.