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Formal Analysis
Motivation:
Output
Recurrent System
Input
Neuron: Dynamics
•Synapse
•Dendrites (Input)
•Cell Body
•Axon (Output)
Output
Input
Equilibrium: Membrane Potential
Threshold
Dendrites: Passive Conductance
Axon: Spikes (Hodgkin Huxley Eqns)
Time
Background: Dynamical Systems Analysis
Phase Space
•Set of all legal states
Dynamics
•Velocity Field
•Flows
•Mapping
Local & Global properties
•Sensitivity to initial conditions
•Fixed points and periodic orbits
Model: Single Neuron
x11
t=0
x12 x13 x14
t=0
x21
t=0
x22
x31 x32
Potential
Function
Each spike represented as: How
long since it departed from soma.
1
1
n1
1
1
2
n2
2
1
m
x21
nm
m
P( x ,..., x , x ,..., x ,...., x ,..., x )
x11
Time
Model: Single Neuron: Potential function
Membrane Potential: Implicit, C  , everywhere bounded function.
P( x1 , x2 ,..., xm )
xi   xi1 , xi2 ,..., xin 
i
Effectiveness of a Spike:
i  1...m, &j  1...ni P
i  1...m, &j  1...ni P
i  1...m, &j  1...ni
dP
xi
P(.)
Threshold:
P(.) = T(.) and
xij
dt
 0 for xij  0
 0 for xij  
j
j
xi  0
0

P(.)
j
xi  
Model: System of Neurons
x11
x12
P1 ( x1 , x2 ,..., xm )
x21
P2 ( x1 , x2 ,..., xm )
x31
P3 ( x1, x2 ,..., xm )
x32
x41
t 0
•Dynamics
•Birth of a spike
•Death of a spike
t
•Point in the Phase-Space
•Configuration of spikes
Model: Single Neuron: Phase-Space
Preliminary: x1 , x2 ,..., xni  0,  i
n
0
Transformation 1:
 zi  e
2 ix i
z1 , z2 ,..., zni  T ni

Transformation 2:
a0 , a1 , ..., ani 1  C ni
 z i  an 1 z n 1  ..  a1 z  a0 
n
i
i
( z  zn ) *.. * ( z  z2 ) * ( z  z1 )
i

0, 
Model: Single Neuron: Phase-Space
Theorem: Phase-Space can be defined formally
Phase-Space for Total Number of Spikes Assigned = 1, 2, & 3.
Model: Single Neuron: Structure of Phase-Space
 Phase-Space for fixed number of Dead spikes:
i
Lσnii
•Dead vs. Live Spikes: Theorem: j  k , i Ljni  i Lkni is an imbedding
 Assign finer topology to
•Phase-Space for n=3
• 1, 2 dead spikes.
i
L0ni
Model: Potential Function: Nature
P’= P•F-1
P’(a1,a2,a3)
P(x1,x2,x3)
F
Analysis:
Continuous Function: C0
Differentiable Function: C1
Continuously Differentiable Function: C2
……
..…..
Smooth Function: C∞
Analysis:
Set of Points
Topological Space:
Metric Space:
Vector Space:
Inner Product Space:
Analysis:
Topological Space:
A topological space is a set X together with a collection T of subsets of X
(i.e., T is a subset of the power set of X) satisfying the following
axioms:
1. The empty set and X are in T.
2. The union of any collection of sets in T is also in T.
3. The intersection of any finite collection of sets in T is also in T.
The set T is called a topology on X. The sets in T are referred to as open
sets, and their complements in X are called closed sets.
A topology specifies "nearness"; an open set is "near" each of its points.
A function between topological spaces is said to be continuous if the
inverse image of every open set is open.
Analysis:
Metric Space:
A metric space M is a set of points with an associated distance function
(also called a metric) d : M × M  R (where R is the set of real
numbers). For all x, y, z in M, this function is required to satisfy the
following conditions:
1. d(x, y) ≥ 0
2. d(x, x) = 0
3. if d(x, y) = 0 then x = y (identity of indiscernibles)
4. d(x, y) = d(y, x) (symmetry)
5. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
Analysis:
Metric Space  Topological Space
In any metric space M we can define the r-neighborhoods as the sets of
the form B(x; r) = {y in M : d(x,y) < r}.
A point x is an interior point of a set E if there exists an r-neighborhood of
x that is a subset of E.
A point x is a limit point of a set E, if every r-neighborhood of x contains
a point y≠x in E.
A set E is open if all points of E are interior points of E.
A set E is closed of all limit points of E belong to E.
Theorem: A set is open if and only if its complement is closed.
Analysis:
Contraction Mapping:
A contraction mapping on a metric space M is a function f from M to
itself, with the property that there is some real number k < 1 such that, for
all x and y in M,
d(f(x),f(y))≤k.d(x,y)
Banach fixed point theorem: Every contraction mapping on a nonempty
complete metric space has a unique fixed point, and that, for any x in M,
the sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point.
A metric space M is said to be complete if every Cauchy sequence of
points in M has a limit in M.
A sequence x1, x2, x3, ... in a metric space (M, d) is called a Cauchy
sequence if for every positive real number r, there is an integer N such
that for all integers m,n> N d(xm, xn) is less than r.
Analysis:
Continuity defined on a metric space, not differentiability.
Attach a Vector Space to every point; Find some way to relate one
Vector space to another.
Cannonical basis….
Analysis:
Field (F,+,*):
1. Closure: For all a,b belonging to F, both a + b and a * b belong to F
2. Associativity of +,*: For all a,b,c in F, a + (b + c) = (a + b) + c and a *
(b * c) = (a * b) * c.
3. Commutativity of +,*: For all a,b in F, a + b = b + a and a * b = b * a.
4. Distributivity: For all a,b,c, in F, a * (b + c) = (a * b) + (a * c).
5. Existence of an additive identity: For all a in F, a + 0 = a.
6. Existence of a multiplicative identity: For all a in F, a * 1 = a.
7. Existence of additive inverses : For all a in F, there exists an element -a
in F, such that a + (-a) = 0.
8. Existence of multiplicative inverses: For every a ≠ 0 in F, there exists
an element a-1 in F, such that a * a-1 = 1.
9. 1≠0
Analysis:
Vector Space:
1. Closure under addition, field multiplication: av+bw belongs to V.
2. Associativity: u+(v+w)= (u+v)+w.
3. Additive identity 0 in V: For all v in V, v+0=v.
4. Additive inverse in V: For all v in V, there exists -v in V: v+(-v)=0.
5. Commutativity: v+w=w+v
6. Associativity: a*(b*v)=(ab)*v.
7. Neutrality of 1 (from Field): 1*v=v.
8. Distributivity w.r.t. vector addition: a*(v+w)=a*v+a*w.
9. Distributivity w.r.t. field addition: (a+b)*v=a*v+b*v.
Analysis:
Inverse Function Theorem:
Let W be an open subset of Rn and F:W Rn be a Cr
mapping.
If there exists an a in W such that DF(a) is non-singular
then:
there exists an open neighborhood U of a in W such that
V=F(U) is open and
F:UV is a Cr diffeomorphism. (F is bijective, both F and
F-1 are continuous and both F and F-1 are Cr mappings.
Analysis:
What is a differential in higher dimensions?
Analysis:
What is a differential in higher dimensions?
Analysis:
How can we prove that the function is even 1-1 given the
dearth of assumptions?
Analysis:
How can we prove that the function is even 1-1 given the
dearth of assumptions?
g(x)=x+(1/f’(a))(y-f(x))
g’(x)=1-f’(x)/f’(a)
Analysis:
Things to ponder:
Why just DF(x) is invertible?
Why not DDF(x), DDDF(x) is invertible etc?
Is D(F-1) = (DF)-1 ?
What about higher order derivatives?