Abstract:
A network of coupled neurons for implementing Non-Lipschitz dynamics for modeling nonlinear processes or conditions comprising: a plurality of neurons, each being configurable in attractor and repulsion modes of operation, and programmable by an external signal; a plurality of synaptic connections for connecting at least a portion of the plurality of neurons for passage of data from one neuron to another; feedback circuitry for incrementing and decrementing an analog voltage output depending upon the output of the synaptic connection; whereby by the circuit solves Non-Lipschitz problems by programmably controlling the attractor and repulsion modes. A method of programming a network for solving Non-Lipschitz problems comprising providing a plurality of neurons, each programmable into a plurality of modes including repulsion and attraction modes; interconnecting the plurality of neurons using synaptic connections; providing feedback to at least one of the neurons; whereby by programming the neurons Non-Lipschitz terminal dynamics can be achieved.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The embodiments herein may be manufactured, used, and/or licensed by or for the United States Government without the payment of royalties. 
    
    
     BACKGROUND OF THE INVENTION 
     Named after Rudolf Lipschitz, the Lipschitz continuity is a form of uniform continuity for functions which are limited to how fast the function can change, i.e., for every pair of points in a graph of a function, the secant of the line segment defined by the points has an absolute value no greater than a definite real number, which is referred to as the Lipschitz Constant. 
     According to Wikipedia, http://en.wikipedia.org/wiki/Lipschitz_continuity, mathematically, a function
 
ƒ: X→Y  
 
is called Lipschitz continuous if there exists a real constant K≧0 such that, for all x 1  and x 2  in X,
 
 d   Y (ƒ( x   1 ),ƒ( x   2 )≦ Kd   X ( x   1   ,x   2 ).
 
where K is referred to as a Lipschitz constant for the function ƒ. The function is Lipschitz continuous if there exists a constant. K≧0 such that, for all x 1 ≠x 2 ,
 
     
       
         
           
             
               
                 
                   ⅆ 
                   Y 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           1 
                         
                         ) 
                       
                     
                     , 
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           2 
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
               
                 
                   ⅆ 
                   X 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       x 
                       1 
                     
                     , 
                     
                       x 
                       2 
                     
                   
                   ) 
                 
               
             
             ≤ 
             
               K 
               . 
             
           
         
       
     
     With regard to Non-Lipschitz mathematics, the publication by Michail Zak and Ronald Meyers entitled “Non-Lipschitz Dynamics Approach to Discrete Event Systems,” Mathematical Modelling and Scientific Computing An International Journal (December 1995) (1995 Zak publication), presents and discusses a mathematical formalism for simulation of discrete event dynamics (DED); a special type of “man-made” systems developed for specific information processing purposes. The main objective of the 1995 Zak publication is to demonstrate that the mathematical formalism for DED can be based upon a terminal model of Newtonian dynamics, which allows one to relax Lipschitz conditions at some discrete points. A broad class of complex dynamical behaviors can be derived from a simple differential equation as described in the 1995 Zak publication and in Michail Zak “Introduction to terminal dynamics,”  Complex Systems  7, 59-87 (1993)1
 
 x=x   1/3  sin ω t , ω=cos  t   (Equation 1A)
 
     In the publication by Michail Zak entitled “Terminal Attractors for Addressable Memory in Neural Networks,” Physics Letters A, Vol. 133, Issues 1-2, pages 18-22 (Oct. 31, 1988) (hereby incorporated by reference), terminal attractors are introduced for an addressable memory in neural networks operating in continuous time. These attractors represent singular solutions of the dynamical system. They intersect (or envelope) the families of regular solutions while each regular solution approaches the terminal attractor in a finite time period. According to the author (Zak), terminal attractors can be incorporated into neural networks such that any desired set of these attractors with prescribed basins is provided by an appropriate selection of the weight matrix. 
     U.S. Pat. No. 5,544,280 to Hua-Kuang Liu, et al. (&#39;280 patent) (hereby incorporated by reference), discloses a unipolar terminal-attractor based neural associative memory (TABAM) system with adaptive threshold for alleged “perfect” convergence. It is noted that an associative memory or content-addressable memory (CAM) is a special type of computer memory in which the user inputs a data word and the memory is searched for storage of the data word. If the data word is located in the CAM, the CAM returns a list of one or more locations or addresses where the data word is located. 
     According to the &#39;280 patent, one of the major applications of neural networks is in the area of associative memory. The avalanche of intensive research interests in neural networks was initiated by the work of J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Nat. Acad. Sci, U.S.A., Vol. 79 p. 2254-258 (1982) (hereby incorporated by reference). U.S. Pat. No. 4,660,106 (in which Hopfield is listed as the inventor) (hereby incorporated by reference) discloses an associative memory modeled with a neural synaptic interconnection matrix and encompasses an interesting computation scheme using recursive, nonlinear thresholding. Further investigation reported that the storage capacity of the Hopfield Model is quite limited due to the number of spurious states and oscillations. In order to alleviate the spurious states problems in the Hopfield model, the concept of terminal attractors was introduced by M. Zak, “Terminal Attractors for Addressable Memory in Neural Networks, Phys. Lett. Vol. A-133, pp. 18-22 (1988)(hereby incorporated by reference). However, the theory of the terminal-attractor based associative neural network model proposed by Zak determines that a new synapse matrix totally different from the Hopfield matrix is needed. This new matrix, which is very complex and time-consuming to compute, was proven to eliminate spurious states, increase the speed of convergence and control the basin of attraction. Zak&#39;s derivation shows that the Hopfield matrix only works if all the stored states in the network are orthogonal. However, since the synapses have changed from those determined by Hebb&#39;s law, Zak&#39;s model is different from the Hopfield model, except for the dynamical iteration of the recall process. According to the &#39;280 patent, the improvement of the storage capacity of the Hopfield model by the terminal attractor cannot be determined based on Zak&#39;s model. The &#39;280 patent discloses a TABAM system which, unlike the complex terminal attractor system of Zak, supra, is not defined by a continuous differential equation and therefore can be readily implemented optically. 
     U.S. Pat. No. 6,188,964 hereby incorporated by reference, purportedly discloses a method for generating residual statics corrections to compensate for surface-consistent static time shifts in stacked seismic traces. The method includes a step of framing the residual static corrections as a global optimization problem in a parameter space. A plurality of parameters are introduced in N-dimensional space, where N is the total number of the sources and receivers. The objective function has a plurality of minimum in the N-dimensional space and at least one of the plurality of minimum is a global minimum. An iteration is performed using a computer; a plurality of pseudo-Lipschitz constants are used to construct a plurality of Pijavski cones to exclude regions on the N-dimensional space where the global minimum is unlikely until a global minimum is substantially reached. See Col. 7, lines 1-10. Using described procedures, it is reported that a reasonably good estimate of the global maximum may be determined. See Col. 9, Lines 45-51. 
     SUMMARY OF THE INVENTION 
     A preferred embodiment of the present invention is directed to the use of analog VLSI technology to implement non-Lipschitz dynamics in networks of coupled neurons for information processing. A component in the analog circuit implementation is an attractor/repeller neuron which faithfully approximates the modulated terminal dynamics {dot over (v)}=s(t)sgn(v)+ε(t). The attractor/repeller neuron may be programmable to become either a terminal attractor or terminal repeller. A preferred embodiment comprises a programmable interconnected network of eight non-Lipschitz neurons implemented in analog VLSI as illustrated in  FIG. 6 . Measurements on the fabricated chip have confirmed the generation of proper terminal dynamics. A circuit board has been developed to interface the chip with instrumentation and to configure the chip in different modes of operation. 
     The method and network use both terminal attractors and terminal repellers in the construction of an analog circuit to achieve a non-Lipschitz dynamics process which can solve important mathematical and physics, problems. The analog circuit uses “noise” in the circuit where the system is near the attractor/repeller points to introduce randomness into the branching of the dynamical paths. If the analog circuit were cooled to low temperature or otherwise implemented in a quantum system the noise there would come from quantum effects. Because of the non-Lipschitz properties, the method can be used to: a) make neural network analog circuits, b) improve the speed of circuits, and c) increase the speed of computer calculations. 
     A preferred embodiment comprises a network of coupled neurons for implementing Non-Lipschitz dynamics for modeling nonlinear processes or conditions comprising: a plurality of attractor/repeller neurons, each of the plurality of neurons being configurable in attractor and repulsion modes of operation, the plurality of attractor/repeller neurons being programmable by an external control signal; a plurality of synaptic connections; at least a portion of the plurality of neurons being interconnected by the synaptic connections for passage of data from one neuron to another; and feedback circuitry for incrementing and decrementing an analog voltage output depending upon the output of the synaptic connection; whereby by the circuit solves Non-Lipschitz problems by programably controlling the attractor and repulsion modes of operation. 
     The invention may be used to solve problems such as the Fokker-Plank equation, Schrödinger equation, and Neural computations. 
     These and other aspects of the embodiments herein will be better appreciated and understood when considered in conjunction with the following description and the accompanying drawings. It should be understood, however, that the following descriptions, while indicating preferred embodiments and numerous specific details thereof, are given by way of illustration and not of limitation. Many changes and modifications may be made within the scope of the embodiments herein without departing from the spirit thereof, and the embodiments herein include all such modifications. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The embodiments herein will be better understood from the following detailed description with reference to the drawings, in which: 
         FIG. 1  illustrates a schematic-diagram of a Terminal Attractor  10 , {dot over (v)}=q(v)+ε(t). 
         FIG. 2  illustrates a schematic diagram of a Terminal Repeller  11 , {dot over (v)}=−q(v)+ε(t). 
         FIG. 3  illustrates a schematic diagram of a Non-Lipschitz Neuron  12 , {dot over (v)}=s(t)q(v)+ε(t). The signal s(t) controls the dynamics through the switches A, R and I: terminal attraction (s=1) when A is selected; terminal repulsion (s=−1) when R is selected; and neither (s=0) when I is selected. 
         FIG. 4  illustrates a schematic diagram of a Synaptic Connectivity, {dot over (x)} i =Σ j T ij q j . 
         FIG. 5  illustrates a schematic diagram of a neural feedback element; i.e., FEEDBACK, ε i =f(x i ). 
         FIG. 6  is a schematic illustration of a preferred embodiment implemented architecture, i.e., the architecture and layout of an NLN (non-Lipschitz neuron) chip containing a coupled array of 8 neurons (center) and 2 individually coupled neurons (bottom). 
         FIG. 7  illustrates sample output waveforms of a single neuron configured alternatingly in terminal attraction and repulsion modes. Top waveform: Synaptic integration x(t); Central waveform: quantization q(t); Bottom waveform: neural state v(t). 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     The embodiments herein and the various features and advantageous details thereof are explained more fully with reference to the non-limiting embodiments that are illustrated in the accompanying drawings and detailed in the following description. Descriptions of well-known components and processing techniques are omitted so as to not unnecessarily obscure the embodiments herein. The examples used herein are intended merely to facilitate an understanding of ways in which the embodiments herein may be practiced and to further enable those of skill in the art to practice the embodiments herein. Accordingly, the examples should not be construed as limiting the scope of the embodiments herein. 
     Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Like numbers refer to like elements throughout. As used herein the term “and/or” includes any and all combinations of one or more of the associated listed items. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the full scope of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein. 
     The paper entitled “Non-Lipschitz Dynamics Approach to Discrete Event Systems,” by Michail Zak and Ronald Meyers (Mathematical Modelling and Scientific Computing An International Journal, December 1995) (hereby incorporated by reference) presents and discusses a mathematical formalism for simulation discrete event dynamics (DED)—a special type of “man-made” systems to serve specific purposes of information processing. The main objective of this work is to demonstrate that the mathematical formalism for DED can be based upon terminal model of Newtonian dynamics, which allows one to relax Lipschitz conditions at some discrete points. A broad class of complex dynamical behaviors can be derived from a simple differential equation, as referenced in Zak, M “Introduction to terminal dynamics,”  Complex Systems  7, 59-87 (1993)(hereby incorporated by reference):
 
 {dot over (x)}=x   1/3  sin ω t , ω=cos  t   (Equation 1A)
 
     The solution to Eq. (1A) can be presented in a closed form. Indeed, assuming that 
     |x|→0 at t=0, one obtains a regular solution: 
                   x   =         ±       (       4     3   ⁢   ω       ⁢     sin   2     ⁢     ω   2     ⁢   t     )       3   2         ⁢           ⁢   if   ⁢           ⁢   x     ≠   0             (     Equation   ⁢           ⁢   2   ⁢   A     )               
and a singular solution (an equilibrium point):
 
 x= 0.  (Equation 3A)
 
     The Lipschitz condition at the equilibrium point x=0 fails since 
     
       
         
           
             
               
                 
                   
                      
                     
                       
                         ⅆ 
                         
                           x 
                           . 
                         
                       
                       
                         ⅆ 
                         x 
                       
                     
                      
                   
                   = 
                   
                     
                       
                         1 
                         3 
                       
                       ⁢ 
                       
                         x 
                         
                           - 
                           
                             2 
                             3 
                           
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                     → 
                     
                       
                         ∞ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         at 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       → 
                       0. 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     As follows from (Equation 2A), two different solutions are possible for “almost the same” initial conditions. The fundamental property of this result is that the divergence of these solutions from x=0 is characterized by an unbounded parameter, U: 
                     σ   =           lim   ⁢               t   →   0       [       1   t     ⁢   ln   ⁢         (       4     3   ⁢   ω       ⁢     sin   2     ⁢     ω   2     ⁢   t     )       3   2         2   ⁢          x   0                ]     =   ∞       ,            x   0          →   0             (     Equation   ⁢           ⁢   5   ⁢   A     )               
where t 0  is an arbitrarily small (but finite) positive quantity. The rate of divergence (5A) can be defined in an arbitrarily small time interval, because the initial infinitesimal distance between the solutions (Equation 2A) becomes finite during the small interval t 0 . Recalling that in the classical case when the Lipschitz condition is satisfied, the distance between two diverging solutions can become finite only at t→0 if initially this distance was infinitesimal.
 
     The solution (Equation 2A) and (Equation 3A) co-exist at t=0, and that is possible because at this point the Lipschitz condition fails (see Eq. 4A). Since: 
                         ⅆ     x   .         ⅆ   x       &gt;     0   ⁢           ⁢   at   ⁢           ⁢        x          ≠   0     ,     t   &gt;   0     ,           (     Equation   ⁢           ⁢   6   ⁢   A     )               
the singular solution (Equation 3A) is unstable, and it departs from rest following Eq. (3A). This solution has two (positive and negative) branches, and each branch can be chosen with the same probability %. It should be noticed that as a result of Equation (4A), the motion of the particle can be initiated by infinitesimal disturbances (that never can occur when the Lipschitz condition is in place since an infinitesimal initial disturbance cannot become finite in finite time).
 
     Strictly speaking, the solution (Equation 2A) is valid only in the time interval 
                     0   ≤   t   ≤       2   ⁢   π     ω       ,           (     Equation   ⁢           ⁢   7   ⁢   A     )               
and at
 
             t   =       2   ⁢   π     ω           
coincides with the singular solution (Equation 3A). For
 
               t   &gt;       2   ⁢   π     ω       ,         
Eq. (2A) becomes unstable, and the motion repeats itself to the accuracy of the sign in Equation (2A).
 
     Hence, the solution performs oscillations with respect to its zero value in such a way that the positive and negative branches of the solution (2A) alternate randomly after each period equal to 
     
       
         
           
             
               
                 2 
                 ⁢ 
                 π 
               
               ω 
             
             . 
           
         
       
     
     Another variable is introduced in Equation (8A):
 
 {dot over (y)}=x , ( y= 0 at  x= 0).  (Equation 8A)
 
     After the first time interval 
     
       
         
           
             t 
             = 
             
               
                 2 
                 ⁢ 
                 π 
               
               ω 
             
           
         
       
     
     
       
         
           
             
               
                 
                   y 
                   = 
                   
                     
                       ± 
                       
                         
                           ∫ 
                           0 
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             ω 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 4 
                                 
                                   3 
                                   ⁢ 
                                   ω 
                                 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               
                                 ω 
                                 2 
                               
                               ⁢ 
                               t 
                             
                             ) 
                           
                           ⁢ 
                           
                             ⅆ 
                             t 
                           
                         
                       
                     
                     = 
                     
                       
                         64 
                         ⁢ 
                         
                           
                             ( 
                             
                               3 
                               ⁢ 
                               ω 
                             
                             ) 
                           
                           
                             - 
                             
                               5 
                               2 
                             
                           
                         
                       
                       = 
                       
                         ± 
                         
                           h 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     After the second time interval 
             t   =       4   ⁢   π     ω             y=±h±h.   (Equation 10A)
 
     Obviously, the variable y performs an unrestricted symmetric random walk: after each time period 
             τ   =       2   ⁢   π     ω           
it changes its value on ±h. The probability f (y, t) is governed by the following difference equation:
 
                       f   ⁡     (     y   ,     t   +       2   ⁢   π     ω         )       =         1   2     ⁢     f   ⁡     (       y   -   h     ,   t     )         +       1   2     ⁢     f   ⁡     (       y   +   h     ,   t     )             ,         ∫     -   ∞     ∞     ⁢       f   ⁡     (     y   ,   t     )       ⁢     ⅆ   y         =   1             (     Equation   ⁢           ⁢   11   ⁢   A     )               
where h is expressed by Eq. (9A).
 
     Equation (11A) defines f as a function of two discrete arguments: 
     
       
         
           
             
               
                 
                   
                     y 
                     = 
                     
                       
                         
                           ± 
                           kh 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       = 
                       
                         l 
                         ⁢ 
                         τ 
                       
                     
                   
                   , 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       τ 
                     
                     = 
                     
                       
                         
                           2 
                           ⁢ 
                           π 
                         
                         ω 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       k 
                     
                   
                   , 
                   
                     l 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   2 
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       etc 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     12 
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     For convenience, for discrete variables y and t the same notations are kept as for their continuous versions. 
     By change of the variables:
 
 z =φ( y ),  y =(φ −1 ( z ),  (Equation 13A)
 
one can obtain a stochastic process with a prescribed probability distribution:
 
                     ψ   ⁡     (     z   ,   t     )       =       f   ⁡     [         φ     -   1       ⁡     (   z   )       ,   t     ]       ⁢              ⅆ     φ     -   1           ⅆ   z            .               (     Equation   ⁢           ⁢   14   ⁢   A     )               
implemented by the dynamical system (Equation 1A), (Equation 8A), and (Equation 13A).
 
     Actually this process represents a piecewise deterministic Markov process with the correlation time τ. However, by introducing a new variable: 
                         u   .     ⁡     (   t   )       =       ∑     q   =   0     n     ⁢       α   q     ⁢     x   ⁡     (     t   -   q   -   r     )             ,       where   ⁢           ⁢     α   q       =   const             (     Equation   ⁢           ⁢   15   ⁢   A     )               
instead of (8A), one arrives at a non-Markov stochastic process with the correlation time (n+1)τ. The deterministic part of the process can be controlled if instead of (Equation 8A) one applies the following change of variables:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         . 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           q 
                           = 
                           0 
                         
                         n 
                       
                       ⁢ 
                       
                         
                           b 
                           q 
                         
                         ⁢ 
                         
                           x 
                           
                             
                               2 
                               ⁢ 
                               sq 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         b 
                         q 
                       
                     
                     = 
                     
                       const 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     In particular, the deterministic part of the process can include a pause it for instance, Equation 14A is reduced to the following:
 
 {dot over (u)} ( t )= x ( t )+ X ( t −τ)  (Equation 17A)
 
     Returning to Equation 1A and assuming that it is driven by a vanishingly small input ε:
 
 {dot over (x)}=x   1/3  sin ω t+ε, ε→ 0  (Equation 18A)
 
     From the viewpoint of information processing, this input can be considered as a message or an event. This message can be ignored when x≠0, or when {dot over (x)}=0, but the system is stable, i.e., x=πω, 2πω, . . . etc. However, it becomes significant during the instants of instability when {dot over (x)}=0, at 
               t   =   0     ,     π     2   ⁢   ω       ,         
. . . etc. Indeed, at these instants, the solution to (18A) would have a choice to be positive or negative if ε=0, (see Equation (2A)).
 
     However, with ε≠0 
                       sgn   ⁢   x     =       sgn   ⁢           ⁢   ε   ⁢           ⁢   at   ⁢           ⁢   t     =   0       ,     π     2   ⁢   ω       ,     …   ⁢           ⁢     etc   .               (     Equation   ⁢           ⁢   19   ⁢   A     )               
i.e. the sign of ε at the critical instances to time (Equation 19A) uniquely defines the evolution of the dynamical system (Equation 18A).
 
     Actually the event ε may represent an output of a microsystem which uniquely controls the behavior of the original dynamical system (Equation 19A). 
     The probability f(y, t), is governed by the following difference equation: 
                     f   ⁡     (     y   ,     t   +       2   ⁢   π     ω         )       =       pf   ⁡     (       y   -   h     ,   t     )       +       (     1   -   p     )     ⁢     f   ⁡     (       y   +   h     ,   t     )                   (     Equation   ⁢           ⁢   20   ⁢   A     )               
where
 
     
       
         
           
             
               
                 
                   p 
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 if 
                                 ⁢ 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                                 ⁢ 
                                 sgn 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ε 
                               
                               = 
                               1 
                             
                           
                         
                         
                           
                             
                               
                                 0 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sgn 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ε 
                               
                               = 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ε 
                               
                               = 
                               0 
                             
                           
                         
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     21 
                     ⁢ 
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
     Actually, the evolution of the probability distribution in Equation 20A is represented by rigid shifts of the initial probability distribution f(y, 0), unless sgn ε=0. 
     The applications of the non-Lipschitz dynamics include stochastic model fitting for identification of physical, biological and social systems, simulation of collective behavior, models of neural intelligence (as discussed in Zak; NI “Introduction to terminal dynamics,”  Complex Systems  7, 59-87 (1993)]; and Zak, M “Physical models of cognition,”  Int. J. of Theoretical Physics  5 (1994), both of which are hereby incorporated by reference). 
     Non-Lipschitz coupled dynamics in neural networks offer an attractive computational paradigm for combining neural information processing with chaos complexity and quantum computation, as discussed in M. Zak, J P Zbilut and R E Meyers,  From Instability to Intelligence , Lecture Notes in Physics 49, Springer-Verlag (1997). A fundamental component in the non-Lipschitz models is the terminal attractor:
 
 {dot over (v)}=q ( v )+ε( t )  (Equation 22A)
 
where the function q(v) is monotonically decreasing and zero at the origin, but with a singularity in the first-order derivative. The singularity at the origin allows the state variable v to reach the stable equilibrium point v=0 in finite time, and remain there indefinitely.
 
     A terminal repeller is obtained by time-reversal of the attractor dynamics:
 
 {dot over (v)}=−q ( v )+ε( t )  (Equation 23A)
 
with an unstable equilibrium point at the origin. The singularity in the first order derivative now allows the state variable v to escape the equilibrium in finite time, even in the absence of noise ε(t). Variable timing in the escape of terminal repellers contribute randomness to an otherwise deterministic system.
 
     For information processing, it is desirable to modulate the terminal dynamics of v(t) with an external signal s(t), that can switch between terminal attraction and repulsion modes of operation:
 
 {dot over (v)}=s ( t ) q ( v )+ε( t )  (Equation 24A)
 
where s(t)&gt;0 for terminal attraction, s(t)&lt;0 for terminal repulsion, and s(t)=0 for stationary dynamics. The timing of the signal s(t) allows to enforce zero initial conditions on the state variable v(t) in attractor mode, for subsequent repulsion as influenced by ε. The critical dependence of the trajectory v(t) on initial conditions in the noise (or signal) ε(t) can be exploited to generate coupled nonlinear chaotic dynamics in a network of neurons. The model of neural feedback can be generally written with activation function
 
ε i   =f ( x   i )  (Equation 25A)
 
and with synaptic coupling dynamics
 
 x   i =Σ j   T   ij   q   j   (Equation 26A)
 
Various instances of the model with different activation functions f(.), synaptic coupling strengths T ij , and modulation sequence s(t), give rise to a vast array of entirely different dynamics that model a variety of physical phenomena from turbulence in fluid mechanics to information processing in biological systems. See, for example, M. Zak, JP Zbilut and R E Meyers,  From Instability to Intelligence , Lecture Notes in Physics 49, Springer-Verlag (1997) (hereby incorporated by reference)
 
Circuit Model
 
     The main difficulty in the circuit implementation of non-Lipschitz terminal attraction and repulsion lies in precise realization of the singularity of the first order derivative in the state variable. Strictly speaking, it is impossible to implement the desired singularity as it would require an amplifier with both infinite gain and infinite bandwidth. A suitable approximation that generates terminal dynamics within the limits of noise in the circuits is desired. 
     Terminal Attractor  10   
     Without loss of generality a binary quantizer element for q(v) is considered:
 
 q ( v )=· sgn ( v )  (Equation 27A)
 
conveniently implemented using a high-gain inverting amplifier  15 , quantizing the analog voltage v to a digital voltage q.  FIG. 1  shows the implementation of the terminal attractor  10  using the quantization model (Equation 27A). Two complementary MOS differential pairs ( 2 ,  3 ) steer a current of polarity controlled by q into a capacitor ( 6 ) on the node v, generating the dynamics (Equation 22A). The temporal scale of the terminal dynamics is given by the capacitance and the tail currents of the differential pairs, set by nMOS and pMOS bias voltages V tn  and V tp , respectively.
 
     Parasitics on the q node effect a delay in the quantization which could potentially give rise to limit cycle oscillations in v(t) around the terminal attractor state. The amplitude of these oscillations can be made arbitrarily small by increasing the dominance of the pole corresponding to the v node. This is accomplished by increasing the capacitance on v, and decreasing the amplitude of the tail currents. It is also possible to internally compensating the amplifier, at the expense of the sharpness of the singularity in the characteristic q(v). 
     Terminal Repeller  12   
     The terminal attractor circuit  10  can be modified to implement terminal repulsion by inclusion of an additional inversion stage between the inverting amplifier output and the current-steering differential pairs. Terminal repeller  12  comprises subcircuits  10 A (also appearing in  FIG. 1) and 11  as shown in  FIG. 2 . The inversion stages are shown in the terminal repeller  12  of  FIG. 2  as two additional inverting amplifiers. The reason for using complementary (pseudo-nMOS and pseudo-pMOS) structures for inverting amplifiers is to provide hysteresis in the positive feedback dynamical response of q(v). The pseudo-nMOS amplifier (top) provides a threshold near the lower supply range, and the threshold of the pseudo-pMOS amplifier (bottom) approaches the higher supply. 
     The delay introduced by the extra stage is not a source of concern as positive feedback excludes the possibility of limit oscillations. More immediately of concern are effects of transistor offset and mismatch in the circuit implementation. These effects can be minimized by reusing the same circuit components for implementing both terminal attraction and repulsion in alternating modes of operation. 
     Non-Lipschitz Neuron  13   
     The non-Lipschitz neuron implements the modulated form (Equation 24A) and in effect combines both terminal attraction and repulsion circuits of  FIGS. 1 and 2  in a single circuit, shown in  FIG. 3 . The modulation signal s(t) takes one of three levels {−1, 0, +1}, supplied in the form of switch control signals R (Repel), I (Inert) and A (Attract), respectively. The signals A and R are complementary (set globally by a single R/overline A control line), and configure the circuit either as the attractor of  FIG. 1 , or the repellor of  FIG. 2 . Signal I breaks either of the two feedback loops and leaves v in a high-impedance (inert) state. 
     By using the same quantizer element in all three operation modes, it is possible to initialize the node voltage vin attraction mode to the same equilibrium value as in repulsion mode. Switching s(t) (R/overline A) then turns the equilibrium point from terminally stable to terminally unstable. The hysteresis [dead zone of feedback from the output q(v)] in repulsion mode allows to integrate the external neural signal ε(t) over time before positive feedback selects a definite direction in the output. For a zero signal component of ε, noise present internally in the circuit will cause the output to randomly decide polarity. 
     Synaptic and Neural Network Dynamics 
     Synaptic connectivity according to Equation (26A) is implemented using the current-inode circuit cell shown in  FIG. 4 . The circuit takes a digital input q j  that selects between one of two current sources T ij   +  (q j =1) and T ij   −  (q j =−1), to contribute a current T ij q j  onto an integration node x i  ( 42 ) Both current sources are bipolar, and are implemented as the difference between positive currents collected on the drain terminals of nMOS  43 ,  43 A and pMOS transistors  44 ,  44 A. The nMOS current sinking transistors  43 ,  43 A are supplied with bias voltages T ij   n+  and T ij   n−  (near GND) on the gate terminals, and similarly the pMOS current sourcing transistors  44 ,  44 A are supplied with bias voltages T ij   p+  and T ij   p−  (near Vdd). The hashmarks shown in  FIG. 4  indicate that the states q j  couple to several x i  (integration nodes  42 ) in the form of several synapses T {ij} . The source terminals are switched by q j  (and  q j   ) rather than gate terminals, to avoid switch injection noise during transients and to allow precise control of very small currents, as needed to attain large dynamic range in neural integration times. 
     Referring now to  FIG. 4 , the values for the currents T ij  may be determined as follows. T ij   n+  onto nMOS  43  generates a current out of x i  (negative into x i ) when q j  is logic high. Conversely, T ij   n−  onto nMOS  43 A generates a current out of x i  (negative into x i ) when q j  is logic low; T ij   p+  onto pMOS  44  generates a current into x i  (positive into x i ) when q j  is logic high; and T ij   p−  onto pMOS  44 A generates a current into x i  (positive into x i ) when q j  is logic low. Therefore, the network implements the following net current into x i , integrated onto node  42  in  FIG. 5 :
 
 Cdx   i   /dt=sum   j (( T   ij   p+   ·T   ij   n+ ) q   j +( T   ij   p−   −T   ij   n− )   q     j )
 
where q j  takes-values 0 (logic low) and 1 (logic high), and  q   j  is its complement. The constant values for the generated currents T ij   n+ , T ij   n− , T ij   p+ , T ij   p−  in this model are controlled by the voltages (and are exponential in the voltages) on these T ij   n+ , T ij   n− , T ij   p+ , T ij   p−  nodes.
 
     In this regard, see G. Cauwenberghs and A. Yariv; “Fault tolerant dynamic multi-level storage in analog VLSI,”  IEEE Trans: Circuits and Systems II: Analog and Digital Signal Processing  41(12), 827-829 (1994) (hereinafter Cauwenberghs &#39;94); and G. Cauwenberghs, “Analog VLSI stochastic perturbative learning architectures,”  Int. J. Analog Integrated Circuits and Signal Processing , 13(1/2), 195-209 (1997), both of which are hereby incorporated by reference. The drain terminals of nMOS  43 ,  43 A, and pMOS  44 ,  44 A transistors connect at an integration node  42 . 
     The purpose of the generation of currents by  43 ,  43 A,  44 , and  44 A is described in Cauwenberghs &#39;94, which describes the implementation of an increment/decrement device in contact with a capacitive storage device similar to capacitor  51  in  FIG. 5 . In this instance, q j  correlates to the binary quantization function described in Cauwenberghs &#39;94. The resulting analog voltage function V i  is the analog level defining the state of the memory, and δ the size of the partial increments, the procedure combines the consecutive steps of binary quantization q j  ( ): R−+{−1, +1}; and Incremental Refresh:
 
 V   i   ≈V   i   +ΣT   ij  
 
Iteration of this procedure yields a stored memory value V i  as described further Cauwenberghs &#39;94. This charge-pump type of implementation of the increment/decrement device in CMOS technology, in contact with a capacitive storage device  51  results in fixed charge increments or decrements on. Capacitive storage device  51  by selectively activating one of four supplied constant currents T ij   n+ , T ij   n− , T ij   p+ , T ij   p−  of sometimes opposite polarity, over a fixed time interval.
 
     Further as to  FIG. 4 , the circuit  40 A is replicated once for each source neuron j. The circuit  40 B is replicated for every synapse Tij (every pair of source neuron j and destination neuron i). Also appearing in  FIG. 4  are high gain inverting amplifiers  46  and capacitors  48 . 
     The neural feedback element is shown in  FIG. 5 . The current Σ j T ij q j  is integrated onto one of the capacitive nodes x i  ( 42 ), and nonlinearly transformed into the output ε i =f(xi). Control of bias voltages V nc , V pc , and determines the shape of the saturating function ƒ(x). Alternatively, digital on/off switching of V nc  and V pc  allows one to implement discrete-time neural feedback when desired. 
     VLSI Implementation and Experimental Results 
       FIGS. 3 ,  4 , and  5  are connected together through the nodes v, q, and x, as labeled in the figures.  FIG. 6  shows how the elements are connected together on the chip in two structures (8 chain-coupled neurons, and 2 cross-coupled neurons, respectively). More specifically,  FIG. 3  shows the circuit diagram of a two-port element with analog input v and digital output q, and control variable s with three programmable states I (s=0), A (s=1) and R (s=−1).  FIG. 1  and  FIG. 2  are for illustration purposes and show the operation of  FIG. 3  when s=1 (terminal attractor mode) and s=−1 (terminal repeller mode), respectively.  FIG. 4  shows the circuit diagram of a synapse, with input q j , and output current x i  at an integration node  42 . The bottom half of  FIG. 4  is replicated once for every source neuron j; the top half is replicated for every synapse T ij  (every pair of source neuron j and destination neuron i).  FIG. 5  shows the circuit diagram of the feedback element, with input current x i , and output voltage v i . The neuron outputs q j  connect through synapses T ij  to other neurons x i  ( FIG. 4 ), through feedback to v i  ( FIG. 5 ), and through the non-Lipschitz neural elements ( FIG. 3 ) back to neural outputs q i  closing the loop. 
       FIG. 6  is a schematic illustration of a preferred embodiment implemented architecture, i.e., the architecture and layout of an NLN (non-Lipschitz neuron) chip. The top of  FIG. 6  shows a network of eight “nearest-neighbor coupled” non-Lipschitz neurons  13 , with identical nearest-neighbor synaptic connections (T 12  and T 21  (double line)), included along with the neuron circuits in each rectangular box  13 ). The synaptic connections T 12  and T 21  are between neighbors in the ring of 8 neurons. Each neuron  13  connects to its upper neighbor neuron  13  with synaptic strength T 12  and to its lower neighbor neuron  13  with synaptic strength T 21 . The multiplexer (MUX) on the left allows one to look at the signal variables of any of the seven left neurons  13 , together with those for the eighth neuron  13  on the right. 
     The lower part of  FIG. 6  illustrates an additional two individual coupled neurons  13 ′ that have been implemented on a 2 mm×2 mm chip in 1.2 μm CMOS technology. T′ 22 , T′ 11 , T′ 12 , T′ 21  in the lower block represent synaptic connections T′ ij  between presynaptic neuron terminals qj and postsynaptic neuron terminals x i , as defined in  FIGS. 4 and 5 , where both indices i and j take values 1 and 2 (there are two neurons). Each of the two neurons  13 ′ has the feedback element of  FIG. 5  (with input current x i ), connecting to the non-Lipschitz element of  FIG. 3 , and the sampler in the bottom of  FIG. 4  (with sampled binary output q i  and its complement  qi ). There are two of these for the two neurons  13 ′. The T ij  elements are positioned in the top part ( 40 B) of  FIG. 4  (with sampled neural states q j  and  qj  as input, and current x i  as output). There are four of these, connecting each neuron to itself and the other neuron. 
     All cells are fully accessible through external inputs and multiplexed outputs. Timing of terminal dynamics is digitally controlled and synaptic connectivity and neural transfer characteristics are programmable through analog bias voltages. The chip tested fully functional, and has been integrated on a PCB board for characterization and experimentation. Example waveforms from one neuron in the chip, configured alternatingly in attractor and repulsion modes of operation, are shown in  FIG. 7 .  FIG. 7  illustrates sample output waveforms of a single neuron configured alternatingly in terminal attraction and repulsion mode. Top waveform of  FIG. 7  illustrates synaptic integration x(t). The central waveform illustrates quantization q(t). The bottom waveform of  FIG. 7  illustrates neural state v(t). 
     As used herein, the terminology “neuron” means an electrically excitable cell or element that processes and transmits information by electrical signals. Each neuron has at least one input and at least one output. 
     As used herein, the terminology “attractor” relates to elements that provide, singular solutions of the dynamical system. They intersect (or envelope) the families of regular solutions while each regular solution approaches the terminal attractor in a finite time period. The methodology may utilize a closed form of the model to analytically reduce the system dynamics onto a stable invariant manifold, onto which empirical data is “attracted.” According to Wikipedia, a trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor). 
     As used herein Tij represents interconnections between the neurons. Iterative adjustments of Tij may be a result of comparison of the net output with known correct answers (supervised learning) or as a result of creating new categories from the correlations of input data when the correct answers are not known (unsupervised learning). 
     The foregoing description of the specific embodiments are intended to reveal the general nature of the embodiments herein that others can, by applying current knowledge, readily modify and/or adapt for various applications such specific embodiments without departing from the generic concept, and, therefore, such adaptations and modifications should and are intended to be comprehended within the meaning and range of equivalents of the disclosed embodiments. It is to be understood that the phraseology or terminology employed herein is for the purpose of description and not of limitation. Therefore, while the embodiments herein have been described in terms of preferred embodiments; those skilled in the art will recognize that the embodiments herein can be practiced with modification within the spirit and scope of the appended claims.