Bifurcation and Multistability in Coupled Neuron Networks with Nonmonotonic Communication

1
Bifurcation and Multistability in Coupled Neuron
Networks with Non-monotonic Communication
Jianfu Ma and Jianhong Wu, Department of
Mathematics and Statistics, York University
A Network of Coupled Two Neurons

• Three fold bifurcations
• a supercritical pitch-fork bifurcation with
  Ws(x,y)y-x, Wc(x,y)yx at the
• curve AB
• a supercritical pitch-fork bifurcation with
  stable manifold Ws(x,y)yx and
• center manifold at the curve CD
• a saddle-node bifurcation with center manifold
  Wc(x,y)y-x and stable manifold
• at the curve EF

• with the norm form
• Simulation in Figure 3 shows,
• When tlt t0, equilibria E4, E5 are asymptotically
  stable.
• When t t0, equilibria lose their stability and
  a supercritical Hopf bifurcation occurs.
• When tgt t0, limit cycle G4 coexists with limit
  cycle G5 in the opposite direction.

A normalized network of two neurons coupled via
delayed feedback with a non-monotonic
activation function

• Effect of time delay on multistability
• The Hopf bifurcation leads to changing the type
  of multistability
• Four stable limit cycles G4, G5, G6, G7 replace
  stable equilibria E4, E5, E6, E7, coexist
• with two other stable equilibria E10, E11.

Motivation Increasing the networks capacity for
memory storage by introducing non-monotonic
activation functions and time lags.

• Multistability
• Two stable equilibria between G and H four
  stable equilibria between H and I six
• equilibria after I.
• Cause of multistability is pitch-fork
  bifurcations and saddle-node bifurcation.

Bifurcation and Multistability whent0 An
equilibrium satisfies Observation (i)
(x,y)(0,0) is an equilibrium. (ii) If (x,y) is
an equilibrium, then so is (y,x). (iii) If (x,y)
is an equilibrium, then so is (-x,-y). Analysis
of the characteristic equation at a given
equilibrium yields Theorem 1 If f(x)f(y)lt a
2, the equilibrium E(x,y) is locally
asymptotically stable if f(x)f(y)gta2, the
equilibrium is unstable. Bifurcation occurs when
f(x)f(y)a2. In the ß1, ß2 plane of the
parameter space, regions of one, three, seven
and eleven equilibria are separated by the
bifurcation sets AB, CD, EF the stability of
equilibria is listed in Table 1 bifurcation
diagram is shown in the middle Figure.
Global Behaviors Butterfly Phenomena
Impact of Time Delay Temporal Pattern
Storage Eigenvalues of the characteristic
equation of the linearized equation are given
by Theorem 2 Equilibria E1, E2, E3, E8, E9,
E10, E11 preserve their stability when tgt0
equilibria E4, E5, E6, E7 also remain stable when
a2lt f(x) f(y) a2. Theorem 3 If f(x) f(y)
lt -a2, we have the following results (i) The
characteristic equation has a pair of purely
imaginary eigenvalues ?i? about equilibria
E4, E5, E6, E7, satisfying (ii) for t ?0, t
0), all eigenvalues of the characteristic
equation have negative real parts and
equilibria are asymptotically stable (iii) When
t t0, except ? i?, all other eigenvalues have
negative real parts (iv) When t?(tj-1, tj), the
characteristic equation has 2j eigenvalues with
positive real parts for j0,1,2,, i.e.,
equilibria become unstable. Because ?i? is a
simple eigenvalue and transversality condition
holds, The system undergoes a supercritical
Hopf bifurcation about equilibria E4, E5, E6, E7
Figure 4 Global dynamical behaviors (butterfly
phenomena)

• Interaction among limit cycles and other
  equilibria gives rise to very interesting global
  behaviors butterfly phenomena and period
  doubling bifurcation
• Amplitude of two limit cycles G4, G5 increases
  as t
• increases, but are separated by the line yx.
• When G4, G5 are tangent to the line yx in
  Figure (c), a
• baby butterfly is developed with two wings and
  the body.

• As delay increases, G4 and G5 cross
  transversally, the trunk of body is developed,
• and a head is developed at the equilibrium E2.
  
• Two butterflies meet at the original point, but
  are separated by the line y-x.
• When two limit cycles G4,5 and G6,7 cross
  transversally each other and connect with the
• body of the other butterfly, two adults are
  finally developed, with head, two wings,
• body and tail.

Figure 2 Region of different equilibria,
bifurcation diagram and phase portrait.

• Discussions
• Multistability lies at the basis of mechanism
  for content-addressable memory and
• retrieval where each equilibrium is identified
  with a static memory, and stable periodic
• solutions are associated with temporally
  patterned spike trains.
• Non-monotonic activation function yields large
  capacity the coexistence of six
• stable equilibria becomes possible.
• A small time delay can be considered as the
  local communication between neurons,
• which does not change the stability of
  equilibria in our neural network.
• The larger time delay can be considered as the
  long range interactions in polysynaptic
• loops of neurons or neuron populations, which
  will change type of multistability related
• from the Hopf bifurcation.

Figure 3 Hopf bifurcation and multistability.