Broca

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Motor cortex
Somatosensory cortex
Sensory associative cortex
Pars opercularis
Visual associative cortex
Brocas area
Visual cortex
Primary Auditory cortex
Wernickes area
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Neurons
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Synapses
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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.
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A single neuron may exhibit complex firing
patterns.
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Network Activity
Uncorrelated activity
Propagating waves
Synchrony
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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?

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How do we model neuronal systems?

1. Single neurons
2. Synaptic connections between neurons
3. Network architecture

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The Neuron
Electrical Signal Action potential that
propagates along axon
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The Hodgkin-Huxley Model
Andrew Huxley
Alan Lloyd Hodgkin
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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
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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.
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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)
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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(1tanh((v-v1)/v2) n?(V)
.5(1tanh((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)
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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.
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Networks
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Synaptic connections
There may be different types of synapses

• excitatory or inhibitory

• activate and/or inactivate at different time
  rates

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

• Oscillatory processes with many time-scales
• Circadian 24 hours
• Slower homeostatic sleep dept
• Internal sleep structure minutes hours
• Neuronal activity milliseconds

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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
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Intracellular aspects of spindling in the
thalamocortical system
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Sleep involves many parts of the brain
Hobson, Nature Reviews Neuroscience 2002
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These sleep rhythms arise from interactions
between cortical neurons and two groups of cells
within the thalamus RE and TC cell.
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Thalamocortical Network
Ctx



RE
TC
-
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Cells behave differently during Spindling and
Delta
TC
RE
Clusters Do not fire every cycle
7-15 Hz Synchrony
Spindle
1 - 4 Hz Synchrony
Slow Rhythm lt 1 Hz
Delta
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• Questions
• How do we model this system?
• What mechanisms underlie these rhythms?
• Transitions between sleep stages?

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BASAL GANGLIA
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BASAL GANGLIA

• Involved in the control of movement
• Dysfunction associated with Parkinsons and
  Huntingtons disease
• Site of surgical procedures
• -- Deep Brain Stimulation (DBS)

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BASAL GANGLIA
dopamine
SNc
Striatum
CTX
GPe
STN
GPi
Thalamus
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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