Transcript Part1

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 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.
A single neuron may exhibit
complex firing patterns.
V
Periodic
spiking
Bursting
oscillation
Network Activity
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Synchrony
Uncorrelated
activity
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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
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
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)
Na+
Na+
K+ +
K
K+
Na+
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
Model for two mutually coupled cells
v1’ = f(v1,w1) – gsyns2(v1 – vsyn)
w1’ = g(v1,w1)
Cell 1
s1’ = a(1-s1)H(v1-q)-bs1
v2’ = f(v2,w2) – gsyns1(v2-vsyn)
w2’ = g(v2,w2)
s2’ = a(1-s2)H(v2-q)-bs2
Synapses may be excitatory or inhibitory
They may turn on or turn off at different rates
Cell 2
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
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
Spindle
Delta
RE
Clusters
Do not fire
every cycle
7-15 Hz
Synchrony
1 - 4 Hz
Synchrony
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
dopamine
SNc
GPe
Striatum
STN
GPi
Excitation
Inhibition
Thalamus
C
T
X
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
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