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Motor cortex. Somatosensory cortex. Sensory associative cortex. Pars opercularis. Visual associative cortex. Broca’s area. Visual cortex. Primary Auditory cortex. Wernicke’s area. Neurons. Synapses. Neurons and synapses. There are about 10 12 neurons in the human brain. - PowerPoint PPT Presentation
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Broca’sarea
Parsopercularis
Motor cortex Somatosensory cortex
Sensory associativecortex
PrimaryAuditory cortex
Wernicke’sarea
Visual associativecortex
Visualcortex
Neurons
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.
Neurons and synapses
What the brain does results from neuronal activity patterns.What the brain does results from neuronal activity patterns.
A single neuron may exhibit complex firing patterns.
VPeriodic spiking
Burstingoscillation
Network Activity
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Synchrony Uncorrelated activity
Propagating waves
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?
1) Single neurons
2) Synaptic connections between neurons
3) Network architecture
The Neuron
Electrical Signal: Action potential that propagates along axon
The Hodgkin-Huxley Model
Alan Lloyd Hodgkin Andrew Huxley
Hodgkin-Huxley Equations
V = Membrane potential
h, m, n = Channel state variables
Model for action potential in the squid giant axon
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)
Some basic biology
Cells have resting potential: potential difference betweeninside 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.
K+K+
K+
Na+
Na+
Na+
The action potential
K+K+
K+
Na+
Na+
Na+
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.
Networks
Synaptic connections
There may be different types of synapses:- excitatory or inhibitory
- activate and/or inactivate at different time rates
v1’ = f(v1,w1) – gsyns2(v1 – vsyn)
w1’ = g(v1,w1)
s1’ = (1-s1)H(v1-)-s1
v2’ = f(v2,w2) – gsyns1(v2-vsyn)
w2’ = g(v2,w2)
s2’ = (1-s2)H(v2-)-s2
v1’ = f(v1,w1) – gsyns2(v1 – vsyn)
w1’ = g(v1,w1)
s1’ = (1-s1)H(v1-)-s1
v2’ = f(v2,w2) – gsyns1(v2-vsyn)
w2’ = g(v2,w2)
s2’ = (1-s2)H(v2-)-s2
Synapses may be excitatory or inhibitory
They may turn on or turn off at different rates
Cell 2
Cell 1
Model for two mutually coupled cells
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
Slow-Wave Activity: -- Spindles: 7 - 15 Hz ; Wax and Wane -- Delta: 1 - 4 Hz -- Slow Osc. .5 - 1 Hz
Stages of sleep form cyclical pattern
Intracellular aspects of spindling in the thalamocortical system
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
TC RE
Spindle
Delta
ClustersDo not fireevery cycle
7-15 HzSynchrony
1 - 4 HzSynchrony
Slow Rhythm < 1 Hz
Questions:
• How do we model this system?
• What mechanisms underlie these rhythms?
• Transitions between sleep stages?
BASAL GANGLIA
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
SNc
Striatum
GPiSTN
GPe
ExcitationInhibition
CTX
Thalamus
dopamine
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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