Multistability for Delayed Neural Networks via Sequential Contracting Jui-Pin Tseng Department of Mathematical Sciences National Chengchi University January 21, 2016 24th Annual Workshop on Differential Equations This is a joint work with Chang-Yuan Cheng (NPTU), Kuang-Hui Lin (NCTU), and Chih-Wen Shih (NCTU). J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 1 / 37
In this talk We explore a variety of multistability scenarios in the general delayed neural network system. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 2 / 37
In this talk We explore a variety of multistability scenarios in the general delayed neural network system. We derive criteria from different geometric configurations which lead to disparate numbers of equilibria. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 2 / 37
In this talk We explore a variety of multistability scenarios in the general delayed neural network system. We derive criteria from different geometric configurations which lead to disparate numbers of equilibria. We introduce a new approach, named sequential contracting, to conclude the global convergence (to multiple equilibrium points) of the system. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 2 / 37
Background: multistability and time delay Multistability is a notion to describe the coexistence of multiple stable equilibria or cycles. - Such dynamics is essential in several applications of neural networks, including pattern recognition and associative memory storage. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 3 / 37
Background: multistability and time delay Multistability is a notion to describe the coexistence of multiple stable equilibria or cycles. - Such dynamics is essential in several applications of neural networks, including pattern recognition and associative memory storage. Time delays are ubiquitous in many natural and artificial systems. - Delays can modify the collective dynamics of neural networks; for example, they can induce oscillation or change the stability of the equilibrium point. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 3 / 37
Background: multistability and time delay Multistability is a notion to describe the coexistence of multiple stable equilibria or cycles. - Such dynamics is essential in several applications of neural networks, including pattern recognition and associative memory storage. Time delays are ubiquitous in many natural and artificial systems. - Delays can modify the collective dynamics of neural networks; for example, they can induce oscillation or change the stability of the equilibrium point. - Taking time delay into account in mathematical models usually increases mathematical technicality. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 3 / 37
Background: model Hopfield-type neural network: n � x i ( t ) = − µ i x i ( t ) + ˙ [ α ij g j ( x j ( t )) + β ij g j ( x j ( t − τ ij ))] + I i , (1) j =1 i = 1 , 2 , · · · , n . µ i > 0, α ij , β ij : connection weights, I i : bias current sources τ ij ≥ 0: time delays, bounded by τ M g j : activation/output function (introduced later) J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 4 / 37
� � � � � � Classes of activation functions Classes A , B , C . We focus on class A . Let ρ i := max {| u i | , | v i |} , g ′ i ( σ i ) = L i J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 5 / 37
Background: the existing works Existence of multiple equilibrium points: - numbers of equilibria are in terms of n -power of the number of saturated (or near saturated) regions in a n -neuron system, e.g. 3 n , (2 r + 1) n , etc. * We can derive the numbers of equilibria which are not in power of n , e.g. 3, 5, 7, for n = 2. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 6 / 37
Background: the existing works Existence of multiple equilibrium points: - numbers of equilibria are in terms of n -power of the number of saturated (or near saturated) regions in a n -neuron system, e.g. 3 n , (2 r + 1) n , etc. * We can derive the numbers of equilibria which are not in power of n , e.g. 3, 5, 7, for n = 2. Stability/convergence of dynamics: - common restriction 1: cooperative ( α ij , β ij ≥ 0, i � = j ) or competitive ( α ij , β ij < 0, i � = j ) (monotone dynamics theory) - common restriction 2: restricted to the class of piecewise-linear activation functions. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 6 / 37
Let us now present our approach to study the existence of equilibrium points for system (1) J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 7 / 37
Existence of equilibria for system (1) Recall system (1): n � x i ( t ) = − µ i x i ( t )+ ˙ [ α ij g j ( x j ( t ))+ β ij g j ( x j ( t − τ ij ))]+ I i , i = 1 , . . . , n . j =1 J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 8 / 37
Existence of equilibria for system (1) Recall system (1): n � x i ( t ) = − µ i x i ( t )+ ˙ [ α ij g j ( x j ( t ))+ β ij g j ( x j ( t − τ ij ))]+ I i , i = 1 , . . . , n . j =1 Consider the stationary equations for (1): n � F i ( x ) := − µ i x i + ( α ij + β ij ) g j ( x j ) + I i = 0 , i = 1 , . . . , n . (2) j =1 J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 8 / 37
Existence of equilibria for system (1) Recall system (1): n � x i ( t ) = − µ i x i ( t )+ ˙ [ α ij g j ( x j ( t ))+ β ij g j ( x j ( t − τ ij ))]+ I i , i = 1 , . . . , n . j =1 Consider the stationary equations for (1): n � F i ( x ) := − µ i x i + ( α ij + β ij ) g j ( x j ) + I i = 0 , i = 1 , . . . , n . (2) j =1 x = ( x 1 , · · · , x n ) is an equilibrium of system (1) if F i ( x ) = 0 , i = 1 , . . . , n . J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 8 / 37
Existence of equilibria for system (1) Recall system (1): n � x i ( t ) = − µ i x i ( t )+ ˙ [ α ij g j ( x j ( t ))+ β ij g j ( x j ( t − τ ij ))]+ I i , i = 1 , . . . , n . j =1 Consider the stationary equations for (1): n � F i ( x ) := − µ i x i + ( α ij + β ij ) g j ( x j ) + I i = 0 , i = 1 , . . . , n . (2) j =1 x = ( x 1 , · · · , x n ) is an equilibrium of system (1) if F i ( x ) = 0 , i = 1 , . . . , n . Our approach combines a geometric formulation on F i ( x ) and the Brouwer’s fixed-point theorem. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 8 / 37
Brouwer’s fixed-point theorem Brouwer’s fixed-point theorem. Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point. J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 9 / 37
Existence of equilibria in system (1) - Idea Locate a region K := K 1 × · · · × K n , with each K i an interval in R , so that for an arbitrary ( ζ 1 , . . . , ζ n ) ∈ K , for every i = 1 , . . . , n , there exists a solution x i ∈ K i to F i ( ζ 1 , . . . , ζ i − 1 , x i , ζ i +1 , . . . , ζ n ) = 0 . J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 10 / 37
Existence of equilibria in system (1) - Idea Locate a region K := K 1 × · · · × K n , with each K i an interval in R , so that for an arbitrary ( ζ 1 , . . . , ζ n ) ∈ K , for every i = 1 , . . . , n , there exists a solution x i ∈ K i to F i ( ζ 1 , . . . , ζ i − 1 , x i , ζ i +1 , . . . , ζ n ) = 0 . Define a continuous mapping Φ : K → K , satisfying Φ( ζ 1 , . . . , ζ n ) = ( x 1 , . . . , x n ) . J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 10 / 37
Existence of equilibria in system (1) - Idea Locate a region K := K 1 × · · · × K n , with each K i an interval in R , so that for an arbitrary ( ζ 1 , . . . , ζ n ) ∈ K , for every i = 1 , . . . , n , there exists a solution x i ∈ K i to F i ( ζ 1 , . . . , ζ i − 1 , x i , ζ i +1 , . . . , ζ n ) = 0 . Define a continuous mapping Φ : K → K , satisfying Φ( ζ 1 , . . . , ζ n ) = ( x 1 , . . . , x n ) . There exists a x = (¯ x 1 , · · · , ¯ x n ), s.t. Φ( x ) = ( x ), i.e., F i ( x ) = 0 , i = 1 , . . . , n J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 10 / 37
Existence of equilibria in system (1) - Idea Locate a region K := K 1 × · · · × K n , with each K i an interval in R , so that for an arbitrary ( ζ 1 , . . . , ζ n ) ∈ K , for every i = 1 , . . . , n , there exists a solution x i ∈ K i to F i ( ζ 1 , . . . , ζ i − 1 , x i , ζ i +1 , . . . , ζ n ) = 0 . Define a continuous mapping Φ : K → K , satisfying Φ( ζ 1 , . . . , ζ n ) = ( x 1 , . . . , x n ) . There exists a x = (¯ x 1 , · · · , ¯ x n ), s.t. Φ( x ) = ( x ), i.e., F i ( x ) = 0 , i = 1 , . . . , n x is an equilibrium of system (1) (in K ). J.P. Tseng (NCCU) Multistability for Delayed Neural Networks January 21, 2016 10 / 37
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