A flexible suite of programs for modelling the cortex

A flexible suite of programs for modelling the cortex with a mean-field scheme A thesis submitted in partial fulfilment of the requirements for the degree of Master of Engineering in Physics and Electronic Engineering at the University of Waikato by Yuan-Kuei (Jay) CHANG 2007 Acknowledgements I would like to use this opportunity to acknowledge all the people that had help me through my masters. Special thanks to my supervisor Dr. Marcus Wilson. Thank you very much for attending to my queries thoroughly with patience. I am sure some of the questions I got for you must sound silly, but you have always made it easy for me to ask them. Your door was always open for me, and the ease for me to trouble you without making prior arrangements really meant a lot to me. Writing a thesis was not an easy task, I am grateful for your guidance throughout the way. Thank you Dr. Alistair Steyn-Ross for your help for programming. It was easier to construct this software system with your support. Thank you Dr. Michael A Cree for assistance with L TEX. The layout of this thesis gets much better with the thesis template and the help you provided. Last but not least, I would like to express my gratitude to my family and all my friends for all your support, without your support and encouragement, completing my masters will be a great hardship for me. Your positive and joyful comments kept me until the end. iii Abstract The cerebral cortex contains many neurons. The neuron is part of the nervous system and it receives and transmits the electrical signals. These signals are significant to a human’s behaviour. Since the neurons are charged, these charges produce electrical fields, so these neural signals can be measured by using scalp electrodes in electroencephalography (EEG). As long as the brain is not dead, the spontaneous activities of neurons will produce a series of EEG signals. There are many models that have been developed for simulating the cortical signal, and mostly each model is focused towards a different purpose or application. Often, a different computer code has to be written for each different application, and this can be inefficient. Therefore, this project aims to develop a software system for simulating cortical signals where the model used for the system can be changed easily. Furthermore, the system is requested to be versatile and easy-to-use for many applications. The developed system is written in MATLAB in response to a user requirement and mostly applies to any model which uses a mean-field approach. Only the specific inputs need to be modified for changing the model. This thesis details how this system is developed. The main limitation of the system is computational resources, much the same as other cortical modelling. However, all the user requirements had been satisfied. The system can simulate the response of the neurons for any condition and generate simulated EEG data to the user. The user can analyze the cortical activities using the standard signal processing techniques such as a power spectrum. This software is very helpful for the research of sleep and anaesthesia. v Contents Acknowledgements Abstract 1 Introduction 1.1 Biological Background . . . . . 1.2 Modeling Approach . . . . . . . 1.3 Uses of the mean-field approach 1.4 Macrocolumn . . . . . . . . . . 1.5 Aim of the Project . . . . . . . 1.6 Thesis Structure . . . . . . . . 2 Requirement 3 Top-level System Design 3.1 Programming Language . . . . . . . . . 3.2 Software Process Models . . . . . . . . . 3.3 System . . . . . . . . . . . . . . . . . . . 3.3.1 Input Function . . . . . . . . . . 3.3.2 Main Function . . . . . . . . . . 3.3.3 Output Function . . . . . . . . . 3.4 Output Processing Function . . . . . . . 3.4.1 Function Power Spectrum . . . 3.4.2 Function Variable Variation iii v 1 1 2 3 4 4 4 7 17 17 17 18 19 23 23 24 24 24 25 25 30 31 35 36 36 38 40 40 41 42 43 43 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Detailed System Design — Simulation 4.1 function Cortex System . . . . . . . . . . . 4.2 function Runge Kutta Integration . . . . 4.3 function find steady states . . . . . . . . 4.4 function steady states delV lambda . . . 4.5 function Steady States stability 14x14 . 4.5.1 Eigenvalue . . . . . . . . . . . . . . . 4.5.2 Procedure . . . . . . . . . . . . . . . . 4.5.3 Testing Mode . . . . . . . . . . . . . . 4.6 function first order derivative . . . . . 4.7 function noise . . . . . . . . . . . . . . . . 4.8 function find trajec . . . . . . . . . . . . 4.9 function perturbation . . . . . . . . . . . 4.10 function Input . . . . . . . . . . . . . . . . 4.11 function data output . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CONTENTS 45 47 47 47 49 53 55 55 55 58 58 58 60 63 67 72 72 73 73 73 77 77 77 77 79 79 79 79 80 80 81 85 87 89 89 91 91 91 94 101 104 4.12 function save output . . . . . . . . . . . . . . . . . . . . . . . . . 5 Detailed System Design — Output Processing 5.1 function Power Spectrum . . . . . . . . . . . . . . . . 5.1.1 Numerical Implementation for Fourier Transform 5.1.2 Programming Implementation . . . . . . . . . . . 5.2 function Variable Variation . . . . . . . . . . . . . . 6 Results and Testing 6.1 Information of the Initial Conditions . . . 6.2 Stability of the selected steady-state . . . 6.3 Simulation . . . . . . . . . . . . . . . . . . 6.4 Output of the simulation . . . . . . . . . . 6.5 Trajectory . . . . . . . . . . . . . . . . . . 6.5.1 No-path: I.trajectory = 0 . . . 6.5.2 User-path: I.trajectory = 2 . . 6.5.3 Isoflurane: I.trajectory = 1 . . 6.6 Testing for Power Spectrum . . . . . . . . 6.6.1 γi = 65s−1 . . . . . . . . . . . . . 6.6.2 γi = 53s−1 . . . . . . . . . . . . . 6.6.3 γi = 46s−1 . . . . . . . . . . . . . 6.6.4 Overall . . . . . . . . . . . . . . . 6.7 Perturbation: I.kicker = 1 . . . . . . . 6.8 Performance of the system: . . . . . . . . 6.8.1 I.Nspace . . . . . . . . . . . . . . 6.8.2 I.saved variable . . . . . . . . . 6.8.3 Tend & deltat . . . . . . . . . . . 6.8.4 t step 4 tplot & t step 4 graph 6.8.5 Numerical Integration . . . . . . . 6.8.6 Stability . . . . . . . . . . . . . . . 6.8.7 movieFlag . . . . . . . . . . . . . . 6.8.8 Elapsed Time . . . . . . . . . . . . 7 Evaluation 8 Future Work and Improvement 9 Conclusion Appendix A: User-Guide A.1 Introduction . . . . . . . . . . . A.2 System Requirement . . . . . . A.3 Input Procedure for Simulation A.3.1 function Input . . . . . A.3.2 Procedure . . . . . . . . A.4 Simulation . . . . . . . . . . . . A.5 Modifying Input Functions . . A.5.1 function find trajec (find trajec.m) . . . . . . . (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 CONTENTS A.5.2 function perturbation (perturbation.m) . . . . . . . . . . . . . . . . . . . . . A.5.3 function noise (noise.m) . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Changing model . . . . . . . . . . . . . . . . . . . . . . . . . A.6.1 function first order derivative (first order derivative.m) . . . . . . . . . . . . . . . . . A.6.2 function find steady states (find steady states.m) . . . . . . . . . . . . . . . . . . A.6.3 function steady states delV lambda (steady states delV lambda.m) . . . . . . . . . . . . . A.6.4 function Steady States stability 14x14 (Steady States stability 14x14.m) . . . . . . . . . . . . A.7 Output (function data output & function save output) . A.8 Data Processing for Analysis . . . . . . . . . . . . . . . . . . A.8.1 function Power Spectrum . . . . . . . . . . . . . . . A.8.2 function Variable Variation . . . . . . . . . . . . Appendix B: User Requirement Appendix C: MATLAB code C.1 Cortex System.m . . . . . . . . C.2 Runge Kutta Integration.m . . C.3 Input.m . . . . . . . . . . . . . C.4 perturbation.m . . . . . . . . . C.5 noise.m . . . . . . . . . . . . . C.6 find trajec.m . . . . . . . . . . C.7 first order derivative.m . . . . . C.8 find steady states.m . . . . . . C.9 Steady States stability 14x14.m C.10 data output.m . . . . . . . . . C.11 save output.m . . . . . . . . . . C.12 Power Spectrum.m . . . . . . . C.13 Variable Variation.m . . . . . . References ix . . . . 105 . . . . 105 . . . . 105 . . . . 107 . . . . 108 . . . . 110 . . . . . . . . . . . . . . . . . . . . 113 114 115 115 118 121 123 123 125 125 127 127 128 129 131 134 137 139 139 145 147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures 1.1 1.2 3.1 4.1 4.2 4.3 4.4 4.5 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 Diagram of neuron structure. Source: Ref. (4) . . . . . . . . . . . . Schematic of a Macrocolumn. Source: Ref. (16) . . . . . . . . . . . . Block diagram of the system . . . . . . . . . . . . . . . . . . . . . . . Structure of the Simulation . . . Flowchart of the main function . Diagram of Steady-State. Source: Damping definition . . . . . . . . Example of Damping . . . . . . . . . . . . . . . . . Modified . . . . . . . . . . . . . . . . from . . . . . . . . . . . . Ref. . . . . . . . . . . . . (15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 19 26 28 32 37 37 50 51 51 54 54 56 57 58 59 61 62 62 64 64 66 66 67 68 68 69 69 70 70 72 73 74 Flowchart of function Power Spectrum . . . . . . . . Example for showing the concept of settling time and Zoom in of Figure 5.2 . . . . . . . . . . . . . . . . . Example for the power spectrum plot . . . . . . . . . Example for the variable variation plot . . . . . . . . . . . . . . . . . division period . . . . . . . . . . . . . . . . . . . . . . . . . . . Value of the defined steady-states. . . . . . . . . . . . . . . . . . . . . Stability of the selected steady-state . . . . . . . . . . . . . . . . . . . Percentage of completion for the simulation. . . . . . . . . . . . . . . Example of the plots for the simulated cortex plane. . . . . . . . . . . Stability Plot of the sleep domain for γi = 15s−1 . Source: Ref. (2) . Plot of variation in the cortical signal with unstable initial condition. Power Spectrum Analysis. . . . . . . . . . . . . . . . . . . . . . . . . Plot of variation in the cortical signal with unstable initial condition. Power Spectrum Analysis. . . . . . . . . . . . . . . . . . . . . . . . . Path of the trajectory in the stability plot. . . . . . . . . . . . . . . . Plot of variation in the cortical signal with user-defined trajectory. . Division 1 of the case in Section 6.5.2. . . . . . . . . . . . . . . . . . Division 11 of the case in Section 6.5.2. . . . . . . . . . . . . . . . . Division 12 of the case in Section 6.5.2. . . . . . . . . . . . . . . . . Division 13 of the case in Section 6.5.2. . . . . . . . . . . . . . . . . Division 14 of the case in Section 6.5.2. . . . . . . . . . . . . . . . . Spectrogram of the user-defined trajectory. . . . . . . . . . . . . . . . Time variation of γi and λi ratios. . . . . . . . . . . . . . . . . . . . Plot of variation in the cortical signal for isoflurane. . . . . . . . . . 3 in 1 power spectra plot. . . . . . . . . . . . . . . . . . . . . . . . . Power Spectrum generated by this system for comparison with case (a) & (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi xii 6.22 Power Spectrum generated by (c) & (d). . . . . . . . . . . . 6.23 Power Spectrum generated by (e) & (f ). . . . . . . . . . . . 6.24 Example of Perturbation. . . A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 this system . . . . . . . this system . . . . . . . . . . . . . . for . . for . . . . LIST OF FIGURES comparison . . . . . . . comparison . . . . . . . . . . . . . . with . . . with . . . . . . case . . . case . . . . . . 75 76 78 Structure of the system . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Example of simulated cortex plane plot . . . . . . . . . . . . . . . . . 102 Percentage of the simulation progress in MATLAB Command Window 103 Example of Workspace with loaded data . . . . . . . . . . . . . . . . 103 Example of Perturbation activated plot . . . . . . . . . . . . . . . . . 106 algorithm for finding the steady-states . . . . . . . . . . . . . . . . . 111 Example of Power Spectrum Plot . . . . . . . . . . . . . . . . . . . . 116 Example of Spectrogram plot . . . . . . . . . . . . . . . . . . . . . . . 117 Example for the plot of signal variation in time and space . . . . . . 119 List of Tables 3.1 4.1 4.2 6.1 8.1 The standard parameters required by the modified Liley model . . . Output Order of the Variables . . . . . . . . . . . . . . . . . . . . . The standard parameters required by the modified Liley model . . . Elapsed Time for each function . . . . . . . . . . . . . . . . . . . . . Elapsed Time for different cases . . . . . . . . . . . . . . . . . . . . . 22 33 44 80 86 A.1 Relationship between the equation and the sub-function of function steady states delV lambda . . . . . . . . . . . . . . . . . . . . . . 113 xiii