Download lectures 5, 6, and 7 () - Stanford Mathematics

Lectures 5,6,7
Ensembles of membrane proteins
as statistical mixed-signal computers
Victor Eliashberg
Consulting professor, Stanford University,
Department of Electrical Engineering
Slide 1
The brain has a very large but rather simple circuitry
The shown cerebellar network has ~1011 granule (Gr) cells and ~2.5 107 Purkinje (Pr) cells. There
are around 105 synapses between T-shaped axons of Gr cells and the dendrites of a single Pr cell.
Pr
Memory is stored in
such matrices
Slide 2
LTM size:
Cerebelum: N=2,5 107 * 105= 2.51012 B= 2.5 TB.
Neocortex: N=1010 * 104= 1014 B= 100 TB.
Simple “3-neuron” associative neural network (WTA.EXE)
addressing by content
S21(I,j)
S21(i,j)
DECODING
Input long-term memory (ILTM)
N1(j)
RANDOM CHOICE
Output long-term memory (OLTM)
ENCODING
retrieval
Slide 3
A functional model of the previous network [7],[8],[11]
(WTA.EXE)
(1)
(2)
(3)
(4)
(5)
Slide 4
Concept of a primitive E-machine
Slide 5
(α< .5)
s(i) >
;
c
Slide 6
Kandel experiments: molecules involved in STM in Aplysia
(E.R. Kandel. In search of memory. 2006, p.233)
Slide 7
Computational machinery of a cell
Membrane
proteins
Membrane
Nucleus
3nm
18nm
It took evolution much longer to create individual cells than to build systems
containing many cells, including the human brain. Different cells differ by
their shape and by the types of membrane proteins.
Slide 8
Protein molecule as a probabilistic molecular
machine (PMM)
i
Slide 9
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Slide 12
Ensemble of PMMs (EPMM)
E-states as occupation
numbers
Slide 13
EPMM as a statistical mixed-signal computer
Slide 14
Ion channel as a PMM
Slide 15
Monte-Carlo simulation of patch clamp
experiments
Slide 16
Two EPMM’s interacting via a) electrical
and b) chemical messages
Slide 17
Spikes produced by an HH-like model with
5-state K+ and Na+ PMM’s. (EPMM.EXE)
Slide 18
The HH gate model
Inside
~18 nm
Outside
+
Na+
+
+
+
Na+
+
+
Cl
Cl -
-
-
-
K+
+
+
K+
+
+
+
uin ~ -64mV
uout =0
a) Potassium channel with 4 n-gates
~ 3nm
Membrane
b) Sodium channel with 3 m-gates and 1 h-gate
Slide 19
Reduced 5-state HH model for potassium channel
Slide 20
Reduced 8-state HH model for sodium channel
Slide 21
The HH mathematical model
(EPMM.EXE)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
NOTE. The HH mathematical model is an approximation of the HH gate model. It
doesn’t follow rigorously from the HH gate model but does produce similar results
Slide 22
A model of sensitization and habituation in a pre-synaptic terminal
subunit of protein kinase A
Slide 23
A PMM implementation of a putative calcium channel with
sensitization and habituation (not a viable biological hypothesis -- just to
demonstrate the possibilities of the EPMM formalism)
Note. The PMM formalism allows one to naturally represent considerably more complex models.
This level of complexity is not available in traditional ANN models.
Slide 24
Ionic currents and membrane potentials
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