Chapter 4. Formal Tools for the Analysis of Brain-Like Structures and Dynamics (1/2) in Creating Brain-Like Intelligence, Sendhoff et al. Course: Robots Learning from Humans Cheolho Han September 25, Biointelligence Laboratory School of Computer Science and Engineering Seoul National University

Contents Introduction Structural Analysis of Networks Dynamical States Conclusion 2

Introduction © 2015, SNU CSE Biointelligence Lab., 3

Introduction Brains and artificial brainlike structures require mathematical tools for the analysis. © 2015, SNU CSE Biointelligence Lab., 4 How information is processed through dynamics Types of dynamics that abstracted networks can support The brain structure abstracted from neuroanatomical results of the arrangement of neurons and the synaptic connections between them 3. Information Processing 2. Dynamical Phenomena 1. Static Structure

Structural Analysis of Networks © 2015, SNU CSE Biointelligence Lab., 5

Structural Analysis of Networks A neurobiological network has properties that distinguish itself from other networks. To look for those properties, spectral analysis has been developed. The spectral density for the diffusion operator L yields characteristic features that distinguish neurobiological networks from other networks. © 2015, SNU CSE Biointelligence Lab., 6

Spectrum of Networks Observing the spectrum of networks, classes of networks can be distinguished. © 2015, SNU CSE Biointelligence Lab., 7 Spectrum of transcription networks Spectrum of neurobiological networks

Dynamical States © 2015, SNU CSE Biointelligence Lab., 8

Why Study Dynamics? The neuronal structure is only the static substrate for the neural dynamics. The relation between dynamic patterns and cognitive processes has not been clearly revealed. Dynamical patterns are where we begin. © 2015, SNU CSE Biointelligence Lab., 9 Time Static Structure Dynamical States

Dynamical Systems When are systems dynamical? When the state changes are not monotonic of the present states Otherwise, the states would just grow or decrease. Systems can be dynamical if the individual elements are dynamical or the elements are connected. © 2015, SNU CSE Biointelligence Lab., 10

Monotonic vs. Non-Monotonic The individual elements can be updated by monotonic or non-monotonic functions. A sigmoid is a monotonic function. The following two functions are non-monotonic. © 2015, SNU CSE Biointelligence Lab., 11

Independent vs. Coupled The updates of the elements can be independent or coupled. The independent update can be done by: The coupled updates can be done by: © 2015, SNU CSE Biointelligence Lab., 12

Case 1 of Dynamical Systems The first case: coupled and monotonic If the strength w ij of the connection from j to i is negative, the connection is inhibitory; if positive, the connection is excitatory. Therefore, we have dynamical systems. © 2015, SNU CSE Biointelligence Lab., 13 Monotonicity MonotonicNon-Monotonic Dependency IndependentX2. Coupled1.3.

Case 2 of Dynamical Systems The second case: independent and non-monotonic The system is dynamic, but it has chaotic behavior. © 2015, SNU CSE Biointelligence Lab., 14 Monotonicity MonotonicNon-Monotonic Dependency IndependentX2. Coupled1.3.

Case 3 of Dynamical Systems The third case: coupled and non-monotonic In this case, the synchronization of chaotic behavior can occur. As the coupling strength  increases, the solution experiences the state changes: desynchronized  synchronized  desynchronized Synchronization: © 2015, SNU CSE Biointelligence Lab., 15 Monotonicity MonotonicNon-Monotonic Dependency IndependentX2 Coupled13

Conclusion © 2015, SNU CSE Biointelligence Lab., 16

Conclusion Some mathematical tools are required to analyze the brain. Spectral analysis is helpful to understand the structure of the neurobiological network. In consideration of the monotonicity of the update function and the dependency of the elements, several models have been suggested. © 2015, SNU CSE Biointelligence Lab., 17

Thank You © 2015, SNU CSE Biointelligence Lab., 18

References 1. Banerjee, A., Jost, J.: Spectral plots and the representation and the interpretation of biological data. Theory Biosci. 126, (2007) 2. Fig Video. © 2015, SNU CSE Biointelligence Lab., 19

Appendix: Chaotic Behavior © 2015, SNU CSE Biointelligence Lab., 20 X(0) = 0.1X(0)=0.01 X(0) = 0.11