Minimax and Hamiltonian Dynamics of Excitatory-Inhibitory Networks
Authors: Tom J. Richardson, J. C. Lagarias, John J. Hopfield H. Sebastian Seung
Published in: Advances in Neural Information Processing Systems 10  (NIPS 1997) (Main Conference Track), 1997

Abstract: A  Lyapunov function  for  excitatory-inhibitory networks  is  constructed.  The construction assumes symmetric interactions within excitatory and  inhibitory  populations  of  neurons,  and  antisymmetric  interactions  be(cid:173) tween  populations.  The  Lyapunov function  yields  sufficient  conditions  for  the  global  asymptotic  stability  of fixed  points.  If these  conditions  are  violated,  limit  cycles  may be stable.  The relations  of the Lyapunov  function  to optimization theory and classical  mechanics are revealed  by  minimax and dissipative Hamiltonian forms  of the network  dynamics. 
The dynamics of a neural network with symmetric interactions provably converges to  fixed  points under very general assumptions[l, 2].  This mathematical result helped  to establish the paradigm of neural computation with fixed  point attractors[3].  But  in reality, interactions between  neurons in the brain are asymmetric.  Furthermore,  the dynamical behaviors seen in the brain are not confined to fixed point attractors,  but also include oscillations and complex nonperiodic behavior.  These other types  of dynamics can be realized by  asymmetric networks, and may be useful for  neural  computation.  For  these reasons, it is  important to understand the global behavior  of asymmetric neural networks.  The interaction  between  an  excitatory neuron  and  an  inhibitory  neuron  is  clearly  asymmetric.  Here we  consider a  class of networks that incorporates this fundamen(cid:173) tal  asymmetry  of the  brain's  microcircuitry.  Networks  of this  class  have  distinct  populations of excitatory and inhibitory neurons,  with  antisymmetric  interactions 
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H.  S.  Seung, T.  1.  Richardson, J.  C.  Lagarias and 1.  1. Hopfield 
between populations and symmetric interactions within each population.  Such net(cid:173) works display  a  rich repertoire of dynamical behaviors including fixed  points, limit  cycles[4,  5]  and traveling waves[6].  After defining the class of excitatory-inhibitory networks, we  introduce a Lyapunov  function  that  establishes  sufficient  conditions  for  the  global  asymptotic  stability  of  fixed  points.  The  generality  of these  conditions  contrasts  with  the  restricted  nature of previous convergence results,  which applied only to linear networks[5]' or  to nonlinear networks with infinitely  fast inhibition[7]. 
The use of the Lyapunov function is illustrated with a competitive or winner-take-all  network, which consists of an excitatory population of neurons with recurrent inhi(cid:173) bition from  a  single neuron[8].  For this network, the sufficient conditions for  global  stability  of fixed  points  also  happen  to  be  necessary  conditions.  In other  words,  we  have proved global stability over the largest possible parameter regime in which  it holds,  demonstrating the power of the Lyapunov function.  There exists  another  parameter regime in which numerical simulations display limit cycle oscillations[7]. 
Similar convergence proofs for other excitatory-inhibitory networks may be obtained  by tedious but straightforward calculations.  All the necessary tools are given in the  first  half of the paper.  But the rest of the paper explains what makes the Lyapunov  function  especially interesting,  beyond the convergence results  it yields:  its  role  in  a  conceptual framework that relates excitatory-inhibitory networks to optimization  theory and classical mechanics. 
The  connection  between  neural  networks  and  optimization[3]  was  established  by  proofs that symmetric networks could find  minima of objective functions[l, 2].  Later  it  was  discovered  that  excitatory-inhibitory  networks  could  perform  the  minimax  computation of finding  saddle points[9,  10,  11],  though no general proof of this was  given at the time.  Our Lyapunov function finally  supplies such a  proof,  and one of  its components is  the objective function of the network's minimax computation. 
Our Lyapunov function can also be obtained by writing the dynamics of excitatory(cid:173) inhibitory  networks  in Hamiltonian form,  with  extra velocity-dependent  terms.  If  these  extra terms  are  dissipative,  then  the  energy  of the system  is  nonincreasing,  and  is  a  Lyapunov  function.  If the  extra terms  are  not  purely  dissipative,  limit  cycles  are  possible.  Previous  Hamiltonian  formalisms  for  neural  networks  made  the  more  restrictive assumption of purely  antisymmetric interactions,  and did  not  include the effect of dissipation[12]. 
This  paper establishes  sufficient  conditions  for  global  asymptotic  stability of fixed  points.  The  problem  of  finding  sufficient  conditions  for  oscillatory  and  chaotic  behavior  remains  open.  The  perspectives  of minimax  and Hamiltonian  dynamics  may help in this task. 
1  EXCITATORY-INHIBITORY NETWORKS 
The dynamics of an excitatory-inhibitory network is  defined  by 
f(u+Ax-By) ,  TxX+X  TyY+y  =  g(v+BTx-Cy).