Dyskretne i niedyskretne
modele sieci neuronów
Winfried Just
Department of Mathematics, Ohio University
Sungwoo Ahn, Xueying Wang
David Terman
Department of Mathematics, Ohio State University
Mathematical Biosciences Institute
A neuronal system consists of 3 components:
1) Intrinsic properties of cells
2) Synaptic connections between cells
1) Network architecture
Each of these involve many parameters and
multiple time scales.
Basic questions:
• Can we classify network behavior?
• Can we design a network that does something of interest?
Outline of the talk
• ODE models of single-cell dynamics
• A small network suggests discrete model
• Definition of discrete dynamics
• Reduction of ODE dynamics to discrete dynamics
• Network connectivity and discrete dynamics
• Relation to other models of discrete dynamics
Single Cell
v’ = f(v,w)
w’ = g(v,w)
A cell may be:
• excitable or
• oscillatory
Variable v measures voltage across membrane.
It changes on fast timescale.
Variable w is called “gating variable” and roughly
measures the proportion of open ion channels.
It changes on slow timescale.
____ v – nullcline
- - - - w - nullcline
Two Mutually Coupled Cells
v1’ = f(v1,w1) – gsyns2(v1 – vsyn)
Synapses may be:
w1’ = g(v1,w1)
• Excitatory or Inhibitory.
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
Depends on vsyn
• Fast or slow.
Depends on a and b
s – fraction of open synaptic channels
H – Heaviside function
gsyn – constant maximal conductance
Sometimes consider indirect synapses:
xi’ = εax(1-xi)H(vi-q) - εbxxi
si’ = a(1-si)H(xi- qx) - bsi
Introduces a delay
in response of synapse
Empirical observations
When the dynamics of this system is simulated on the computer,
one observes rather sharply defined episodes of roughly equal lengths
during which groups of cells fire (reside on the right branch of the v-nullcline)
together, while other cells rest (reside on the left branch of the v-nullcline).
Membership in these groups may change from episode to episode;
a phenomenon that is called dynamic clustering.
Experimental studies of actual neuronal networks, such as the olfactory bulb
in insects or the thalamic cells involved in sleep rhythms appear to show
similar patterns.
This suggests that one could attempt to reduce the ODE dynamics to a
simpler discrete model and study the properties of the discrete model
instead.
Reduction to discrete dynamics
(1,6)
(4,5)
(2,3,7)
Transient: linear
sequence of activation
(1,5,6)
(2,4,7)
Period: stable, cyclic
sequence of activation
Assume: A cell does
not fire in consecutive
episodes
(3,6)
(1,4,5)
Network
Architecture
Some other solutions
(1,6)
(1,3,7)
(1,2,5)
(4,5)
(4,5,6)
(4,6,7)
(2,3,7)
(2,3,5)
(1,5,6)
(1,6,7)
(2,4,7)
(3,4,5)
(3,6)
(1,2,7)
(1,4,5)
(3,4,5,6)
Different transient
Different attractor
Different transient
Same attractor
What is the state transition graph of the dynamics?
How many attractors and transients are there?
2
7
6
5
4
3
1
Network architecture
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Remarks
1) We have assumed that refractory period = 1
If a cell fires then it must wait one episode before
it can fire again.
2) We have assumed that threshold = 1
If a cell is ready to fire, then it will fire if it received input
from at least one other active cell.
For now, we assume that:
• refractory period of every cell = p
• threshold for every cell = 1
Discrete Dynamics
Start with a directed graph D = [ VD, AD ] and integer p.
2
7
6
5
4
3
1
Discrete Dynamics
Start with a directed graph D = [ VD, AD ] and integer p.
A state s(k) at the discrete time k is a vector:
s(k) = [s1(k), …., sn(k)] where
si(k)  {0, 1, … ,p} for each i. (n = # cells)
The state si(k) = 0 means neuron i fires at time k.
Dynamics on the discrete network:
• If si(k) < p, then si(k+1) = si(k)+1
• If si(k) = p, and there exists a j with sj(k)=0 and
<j,i>  AD, then si(k+1) = 0.
• If si(k) = p, and there is no j with sj(k)=0 and
<j,i>  AD, then si(k+1) = p.
Two Issues
1) When can we reduce the differential equations model
to the discrete model?
2) What can we prove about the discrete model?
In particular, how does the network connectivity
influence the discrete dynamics?
Reducing the neuronal model to discrete dynamics
Given integers n (size of network) and p (refractory period),
can we choose intrinsic and synaptic parameters so that
for any network architecture, every orbit of the discrete
model can be realized by a stable solution of the
neuronal model?
Answer:
No
- for purely inhibitory networks.
Yes - for excitatory-inhibitory networks.
Post-inhibitory rebound
We will consider networks of neurons in which the
w-nullcline intersects the left branch of the v-nullcline(s).
If such a neuron receives excitatory input, the v-nullcline
moves up, if it receives inhibitory input, the v-nullcline
moves down.
If two such neurons are coupled by inhibitory synapses,
the resulting dynamics is known under the name postinhibitory rebound.
Purely Inhibitory Network
1
2
C(1)
g=0
C(0)
cell 1
3
4
cell 2
C(0)
cell 3
(1,2)
C(2)
cell 4
(3,4)
C(1)
Note that the distance between cells
within each cluster increases.
Excitatory-Inhibitory Networks
E-cells
excitation
inhibition
I-cells
Formally reduce E-I network  purely inhibitory network
E-cells
excitation
inhibition
I-cells
E-cell fires
I-cells fire
E-cells fire
due to rebound
We can then define a graph on the set of E-cells
and define discrete dynamics as before.
More precisely:
The vertex set of the digraph consists of the numbers of
all E-neurons. An arc <i,j> is included in the digraph if
and only if there is some I-neuron x that may receive
excitatory input from i and may send inhibitory input to j.
Rigorously reducing E-I networks to discrete dynamics
Assume:
• All-to-all coupling among I-cells
• Inhibitory synapses are indirect (slow)
• Suitable functions f and g
• The ODE dynamics is assumed to be
the dynamics on the slow timescale;
all trajectories move along the v-nullclines;
jumps are instantaneous
Discrete vs. ODE models
Consider any such network with any fixed architecture and fix p,
the refractory period. We can then define both the continuous
neuronal and discrete models. Let P(0) be any state of the
discrete model. We then wish to show that there exists a solution
of the neuronal system in which different subsets of cells take turns
jumping up to the active phase. The active cells during each
subsequent episode are precisely those determined by the discrete
orbit of P(0), and this exact correspondence to the discrete dynamics
remains valid throughout the trajectory of the initial state. We will say
that such a solution realizes the orbit predicted by the discrete
model. This solution will be stable in the sense that there is a
neighborhood of the initial state such that every trajectory that
starts in this neighborhood realizes the same discrete orbit.
Main Theorem (Terman, Ahn, Wang, Just;
Physica D, 237(3) (2008))
Suppose a discrete model defined by a digraph is given. Then there
re are intervals for the choice of the intrinsic parameters of the cells and
the synaptic parameters in the ODE model so that:
1. Every orbit of the discrete model is realized by a stable
solution of the differential equations model.
2. Every solution of the differential equations model eventually
realizes a periodic orbit of the discrete model. That is, if
X(t) is any solution of the differential equations model, then
there exists T > 0 such that the solution {X(t) : t > T } realizes a
periodic orbit or a steady state of the discrete model.
Strategy
Suppose we are given an E-I network.
Let s(0) be any initial state of the discrete system.
We wish to choose initial positions of the E- and I- cells so that the E-I
network produces firing patterns as predicted by the discrete system.
g=0
E-cells
We construct disjoint intervals Jk, k = 0,…,p, so that:
Let s(0) = (s1, ….., sn). Consider E-cells, (vi,wi).
Assume: If si(0) = k, then wi(0)  Jk.
Then:  T* > 0 such that if si(1) = k, then wi(T*)  Jk.
The only E-cells that fire for t  [0,T*] are those with si(0) = 0.
J1
J1
J2
J2
J0
p=2
J0
Generalized Discrete Dynamics
Start with a directed graph D = [ VD, AD ] and vectors of
integers p = [p1, … , pn] and th = [th1, … , thn].
A state s(k) at the discrete time k is a vector:
s(k) = [s1(k), …., sn(k)] where
si(k)  {0, 1, … ,pi} for each i. (n = # cells)
The state si(k) = 0 means neuron i fires at time k.
Dynamics on the discrete network:
• If si(k) < p, then si(k+1) = si(k)+1
• If si(k) = pi, and there exists at least thi nodes j
with sj(k)=0 and <j,i>  AD, then si(k+1) = 0.
• If si(k) = pi, and there are fewer than thi nodes j
with sj(k)=0 and <j,i>  AD, then si(k+1) = pi.
Rigorous analysis of discrete model
Start with a directed graph D = [ VD, AD ]
pi = refractory period of neuron i
thi = threshold of neuron i
n = # of vertices.
How does the expected dynamics of the discrete model
depend on the density of connections?
We will study this question by considering random initial
states in random digraphs with a specified connection
probability.
Some Definitions
Let L = {s(1), …., s(k)} be an attractor.
Act(L) = { i: si(t) = 0 for some t}
(the active set of L)
L is fully active if Act(L) = [n] = {1, … , n}
L is a minimal attractor if Act(L)  Ø and, for each i  Act(L),
si(0), …., si(k) cycle through 0, 1, …, pi.
Let:
MA = {states that belong to a minimal attractor}
FAMA = {states that belong to a fully active minimal attractor}
Consider random digraphs:
(n) = probability <i,j>  AD for given <i,j>.
Theorem: Assume that pi and thi are bounded
independent of n.
(i) If (n) = (ln n / n), then as n  , with probability
one, |FAMA|/|states|  1.
(ii) If (n) = o(ln n / n), then as n  , with probability
one, |MA|/|states|  0.
A phase transition occurs when (n) ~ ln n / n.
Just, Ahn, Terman; Physica D 237(24) (2008)
Autonomous sets
Definition: Let s(0) = [s1(0), …., sn(0)] be a state.
We say A  VD is autonomous for s(0) if for every i  A,
si(t) is minimally cycling (that is, si(0), si(1), …, si(t)
cycles through {0, …., pi}) in the discrete system that is
obtained by restricting the nodes of the system to VD.
Example: Active sets of minimal attractors are
autonomous. Note that the dynamics on an autonomous
set does not depend on the states of the remainder of
the nodes.
The following result suggests that there exists another
phase transition ~ C/n.
Theorem: Assume that each pi < p and thi < th. Fix  
(0,1). Then  C(p, th, ) such that if (n) > C/n, then with
probability tending to one as n  , a randomly
chosen state will have an autonomous set of size at least
n. In particular, most states have a large set of minimally
cycling nodes.
Just, Ahn, Terman; Physica D 237(24) (2008)
Numerical explorations
Current work in progress
When the connection probability is ~ 1/n, another phase
transition occurs for the case when all refractory periods and all
firing thresholds are 1. Below this phase transition, with high
probability the basin of attraction of the steady state [1, … , 1]
becomes the whole state space. We are investigating what
happens for connection probabilities slightly above this phase
transition. Theoretical results predict longer transients near the
phase transition, and this is what we are seeing in simulations.
One question we are interested in is whether chaotic dynamics
would be generic for connection probabilities in a critical range.
Other ongoing research
• Can we generalize our first theorem to architectures
where the connections between the I-cells are
somewhat random rather than all-to-all?
• How to incorporate learning and processing of inputs
into this model?
• Can we obtain analogous results for networks based
on different single-cell ODE dynamics?
Hopfield Networks
• Networks are modeled as digraphs with weighted
arcs; weights may be positive or negative
• Each neuron has a firing threshold thi
• At each step, neurons are in state zero or one
• The successor state of a given state is determined by
summing the weights of all incoming arcs that
originate at neurons that are in state one. If this
weight exceeds thi, the neuron goes into state one
(fires), otherwise it goes into state zero
Hopfield vs. Terman Networks
• Hopfield networks don’t model refractory periods
• Terman networks don’t allow negatively weighted
arcs
• For refractory period p = 1, both kinds are examples
of Boolean networks
• Dynamics of random Hopfield networks tends to
become more chaotic as connectivity increases
• Random Terman networks may allow chaotic
dynamics only for narrow range of connectivity
Why am I interested in this?
• My major interests are centered around models of
gene regulatory networks.
• These can be modeled with ODE systems; but
Boolean and other discrete models are also being
studied in the literature, with the (generally)
unrealistic assumption of synchronous updating.
• Question: Under which conditions can we prove a
correspondence between discrete and continuous
models of gene regulatory networks as in our first
theorem?