Motor cortex Somatosensory cortex Sensory associative cortex Pars opercularis Visual associative cortex Broca’s area Visual cortex Primary Auditory cortex Wernicke’s area
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.
V Periodic spiking Bursting oscillation A single neuron may exhibit complex firing patterns.
Network Activity Uncorrelated activity Propagating waves Synchrony
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?
How do we model neuronal systems? Single neurons Synaptic connections between neurons Network architecture
The Neuron Electrical Signal: Action potential that propagates along axon
The Hodgkin-Huxley Model Andrew Huxley Alan Lloyd Hodgkin
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
+ 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.
+ 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)
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)
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.
Synaptic connections There may be different types of synapses: • excitatory or inhibitory • activate and/or inactivate at different time rates
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
Network Architecture Example: excitatory-inhibitory network Note: There are many different types of connectivities: -- Sparse, global, random, structures, …
Sleep Oscillatory processes with many time-scales: Circadian: 24 hours Slower: homeostatic sleep dept Internal sleep structure: minutes – hours Neuronal activity: milliseconds
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
Sleep involves many parts of the brain Hobson, Nature Reviews Neuroscience 2002
These sleep rhythms arise from interactions between cortical neurons and two groups of cells within the thalamus: RE and TC cell.
Thalamocortical Network Ctx + + + RE TC -
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
Questions: • How do we model this system? • What mechanisms underlie these rhythms? • Transitions between sleep stages?
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)
BASAL GANGLIA Excitation Inhibition dopamine SNc Striatum CTX GPe STN GPi Thalamus
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