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Broca’s area

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  1. Motor cortex Somatosensory cortex Sensory associative cortex Pars opercularis Visual associative cortex Broca’s area Visual cortex Primary Auditory cortex Wernicke’s area

  2. Neurons

  3. Synapses

  4. Neurons and synapses • There are about 1012 neurons in the human brain. • Neurons generate electrical signals (action potentials). • Neurons communicate with each other at synapses. • There are about 1015 synaptic connections. What the brain does results from neuronal activity patterns.

  5. V Periodic spiking Bursting oscillation A single neuron may exhibit complex firing patterns.

  6. Network Activity Uncorrelated activity Propagating waves Synchrony

  7. Mathematical Challenges • How should one model neuronal networks? • What types of activity patterns emerge in a model? • How does these patterns change wrt parameters? • How can we mathematically analyze the solutions? • How does the brain use this information?

  8. How do we model neuronal systems? Single neurons Synaptic connections between neurons Network architecture

  9. The Neuron Electrical Signal: Action potential that propagates along axon

  10. The Hodgkin-Huxley Model Andrew Huxley Alan Lloyd Hodgkin

  11. Hodgkin-Huxley Equations CVt = DVxx - gNam3h(V-Ena) - gKn4(V-EK) - gL(V-EL) mt = (m(V) - m) / m(V) ht = (h(V) - h) / h(V) nt = (n(V) - n) / n(V) V = Membrane potential h, m, n = Channel state variables Model for action potential in the squid giant axon

  12. + Na + + Na K + K + K + Na Some basic biology Cells have resting potential: potential difference between inside and outside of cell Resting potential maintained by concentration differences of ions inside and outside of cell There are channels in membrane selective to different ions. Channels may be open or closed. Membrane potential changes as ions flow into or out of cell.

  13. + Na + + Na K + K + K + Na The action potential CVt = -gNam3h(V-Ena) - gKn4(V-EK) - gL(V-EL) mt = (m(V) - m) / m(V) ht = (h(V) - h) / h(V) nt = (n(V) - n) / n(V)

  14. The Morris-Lecar equations CVt = -gCa m(V) (V-ECa) - gKn(V-EK) - gL(V-EL) + Iapp nt = (n(V) - n) / n(V) m(V) = .5(1+tanh((v-v1)/v2) n(V) = .5(1+tanh((v-v3)/v4) n(V) = 1/cosh((v-v3)/2v4) We will write this system as: V’ = f(V,n) + Iapp n’ = g(V,n)

  15. Class I: (SNIC) Axons have sharp thresholds, can have long to firing, and can fire at arbitrarily low frequencies Class II: (Hopf) Axons have variable thresholds, short latency and a positive frequency.

  16. Networks

  17. Synaptic connections There may be different types of synapses: • excitatory or inhibitory • activate and/or inactivate at different time rates

  18. Model for two mutually coupled cells v1’ = f(v1,w1) – gsyns2(v1 – vsyn) w1’ = eg(v1,w1) s1’ = a(1-s1)H(v1-q)-bs1 v2’ = f(v2,w2) – gsyns1(v2-vsyn) w2’ = eg(v2,w2) s2’ = a(1-s2)H(v2-q)-bs2 Cell 1 Cell 2 Synapses may be excitatory or inhibitory They may turn on or turn off at different rates

  19. Network Architecture Example: excitatory-inhibitory network Note: There are many different types of connectivities: -- Sparse, global, random, structures, …

  20. Sleep Oscillatory processes with many time-scales: Circadian: 24 hours Slower: homeostatic sleep dept Internal sleep structure: minutes – hours Neuronal activity: milliseconds

  21. Stages of sleep form cyclical pattern Slow-Wave Activity: -- Spindles: 7 - 15 Hz ; Wax and Wane -- Delta: 1 - 4 Hz -- Slow Osc. .5 - 1 Hz

  22. Intracellular aspects of spindling in the thalamocortical system

  23. Sleep involves many parts of the brain Hobson, Nature Reviews Neuroscience 2002

  24. These sleep rhythms arise from interactions between cortical neurons and two groups of cells within the thalamus: RE and TC cell.

  25. Thalamocortical Network Ctx + + + RE TC -

  26. Cells behave differently during Spindling and Delta RE TC Clusters Do not fire every cycle 7-15 Hz Synchrony Spindle 1 - 4 Hz Synchrony Slow Rhythm < 1 Hz Delta

  27. Questions: • How do we model this system? • What mechanisms underlie these rhythms? • Transitions between sleep stages?

  28. BASAL GANGLIA

  29. BASAL GANGLIA • Involved in the control of movement • Dysfunction associated with Parkinson’s and Huntington’s disease • Site of surgical procedures -- Deep Brain Stimulation (DBS)

  30. BASAL GANGLIA Excitation Inhibition dopamine SNc Striatum CTX GPe STN GPi Thalamus

  31. Motivation of Computational Study • Explain changes in firing patterns within the basal ganglia • During PD, neurons display: • Increased synchrony • Increased bursting activity • Mechanism underlying DBS mysterious