No subject

approach for the analysis of the Linsker's unsupervised Hebbian learning 
network.   The behavior of  this model  is determined by  the underlying 
nonlinear  dynamics  that  are  parameterized  by  a set  of  parameters 
originating from the Hebbian rule and the arbor density of the synapses.   
These  parameters  determine  the presence  or  absence  of  a  specific 
receptive  field  (also  referred  to  as  a  connection  pattern)  as a 
saturated fixed point attractor  of  the  model.    In  this  paper,  we 
perform a qualitative analysis of the underlying nonlinear dynamics over 
the parameter space,  determine  the effects of the system parameters on 
the emergence  of  various  receptive fields,  and  provide  a  rigorous 
criterion  for the parameter regime  in which  the network will have the 
potential  to  develop  a specially designated  connection pattern.   In 
particular, this approach analytically demonstrates, for the first time,   
the crucial role  played by  the synaptic arbor density.    For example, 
our analytic predictions indicate that  no structured connection pattern  
can emerge  in  a Linsker's network  that is fully feedforward connected   
without localized synaptic arbor density.   Our general theorems lead to  
a complete and precise picture of the parameter space that  defines  the 
relationships  between  the different sets of  system parameters and the 
corresponding  fixed point attractors,  and  yield  a method  to predict  
whether  a given connection pattern  will emerge  under  a given set  of 
parameters  without running a numerical simulation of the model.     The 
theoretical results  are corroborated by our examples (including center-
surround  and  certain  oriented  receptive  fields),   and  match   key  
observations reported in Linsker's  numerical simulation.   The rigorous 
approach presented here  provides  a unified treatment  of many  diverse  
problems about the dynamical mechanism of a class of models that use the 
limiter function  (also  referred to  as  the piecewise linear sigmoidal 
function) as the constraint limiting the size of the weight or the state 
variables,  and  applies not only to the Linsker's network  but also  to 
other    learning    or    retrieval    models     of     this    class.

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Key Words:  Unsupervised  Hebbian learning,  Network  self-organization, 
         Linsker's  developmental  model,   Brain-State-in-a-Box  model, 
         Ontogenesis of primary visual system, Afferent receptive field, 
         Synaptic  arbor  density,   Correlations,    Limiter  function,   
         Nonlinear dynamics,   Qualitative  analysis,   Parameter space,
         Coexistence   of   attractors,    Fixed   point,     Stability.

------------------------------------------------------------------------

Contents:

{1} Introduction
 {1.1} Formulation Of The Linsker's Developmental Model
 {1.2} Qualitative Analysis Of Nonlinear System  And  Afferent Receptive 
       Fields
 {1.3} Summary Of Our Approach

{2} General Theorems About Fixed Points And Their Stability

{3} The  Criterion  For  The  Division  Of  Parameter  Regimes  For  The 
    Occurrence Of Attractors
 {3.1} The  Necessary  And  Sufficient  Condition  For  The Emergence Of 
       Afferent Receptive Fields
 {3.2} The General Principal Parameter Regimes

{4} The Afferent Receptive Fields In The First Three Layers
 {4.1} Description Of The First Three Layers Of The Linsker's Network
 {4.2} Development Of Connections Between Layers A And B
 {4.3} Analytic Studies Of Synaptic Density Functions' Influences In The 
       First Three Layers
 {4.4} Examples Of Structured Afferent Receptive Fields Between Layers B 
       And C

{5} Concluding Remarks
 {5.1} Synaptic Arbor Density Function
 {5.2} The Linsker's Network And The Brain-State-in-a-Box Model
 {5.3} Dynamics With Limiter Function
 {5.4} Intralayer Interaction And Biological Discussion

References
Appendix A: On the Continuous Version of the Linsker's Model
Appendix B: Examples  of  Structured  Afferent Receptive Fields  between 
            Layers B and C of the Linsker's Network

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