graph theory approaches to neural structures and dynamics

CND Lab Memphis & UCB

WCCI/IJCNN2016 Tutorial July 24, 2016, Vancouver, Canada

Graph Theory Approaches to Neural Structures and Dynamics

- Models and Applications

Robert Kozma Department of Mathematics University of Memphis Memphis, TN, USA

In collaboration with Walter Freeman, UC Berkeley

CLION Graph Theory for Neural Structures & Dynamics

Walter Freeman (1927-2016)


Collaborations / Credits

•  NSF/DMS CRCNS “Computational Neuroscience of Strategy Change in Cognitive Biological and Technical Systems,” US-German Collaboration

•  DARPA DSO Physical Intelligence Research Thrust, Phase II “Intentional Action-Perception through Random Graph Theory”

•  AFOSR Math & Cognition Program “Emergence of Near-Critical Regularity in Brain Network Dynamics”

•  AFRL WPAFB Human Effectiveness “Robust Decision Making for Improved Assurance with EEG,”

•  NASA JPL Robotics Laboratory “Dynamical Approach to Behavior-Based Robot Control & Autonomy,”

•  MAS Millennium Institute of Astrophysics University of Chile, Santiago, Chile


Outline I.  Review of experiments on brain dynamics

•  Single cell and population measurements •  Noninvasive techniques, fMRI, MEG, scalp EEG •  Description of the connectome

II.  Random graph theory basics •  Erdos-Renyi, Strogatz-Watts, Albert-Barabasi models •  Scale-free structure and dynamics, transients, black swans and dragon kings •  Percolation and characterization of phase transitions

III.  Computational approaches to large-scale brain networks •  Lattice models, hierarchy of models •  Neuropercolation approach •  Implementation of massive computational models.

IV.  Critical behavior in neural and cognitive networks •  Phase transitions in cognitive processing, •  Interpretation of phase transitions as “aha” moments, •  Neural correlates of higher cognition and consciousness

V.  Applications in engineering •  Advanced Brain Computer Interface techniques, •  Brain connection diseases, support quality of life elderly and disabled.


How brains create knowledge from information

①  • Action-perception cycle of intentional behavior ②  • State variables: macroscopic vs. microscopic ③  • Gaussianity: power-law, scale-free activity ④  • Linearity: additivity and proportionality ⑤  • Stability: ‘spontaneous’ background activity ⑥  • Oscillations: mechanism in negative feedback ⑦  • Criticality: Cortical phase transitions

I Biological Foundations of Modeling Cortex


1.  Components of Intentionality - Limbic System

W J Freeman "How Brains Make Up Their Minds” (Columbia U.P., 2001)

Salamander – olfactory dominance


Hedgehog olfactory system (Adrian, 1950) Airflow EEG Local In response to odor stimulus à Phase transition

Frequency decreases - Magnitude increases

Input-Induced Destabilization of the Cortex


7

EEG, Cat hungry > then satiated

Pial surface Bipolar Depth


Dynamic Systems Approach to Cortical Neurodynamics (Freeman/Skarda’87)

•  The system maintains a state space dominated by a high-dimensional, flexible, evolving attractor landscape.

•  Input is by waves at the mesoscopic level from cortices that overlap but need not be synchronous.

•  Operation is by global phase transitions induced aperiodically by spatial integration of the wave packets. The transitions lead to hemisphere-wide spatial amplitude modulation patterns.


Strong Dynamic Hypothesis (SDH) to Cognition & Intelligence

SDH: Aperiodic (chaotic) dynamics is necessary condition of intelligence in animals and animats (with Harter/Kozma)

Relevance: itinerancy and transitions in brains/cognition at various levels:

EEG – Intermittent desynchronization of hemishpere activity (Freeman) Behavior/motor action – Metastability in cognitive proce (Haken/Kelso) Consciousness – Global Workspace Theory & conscious broadcast (Baars) Brain theories – 2nd and 3rd Generation (Buzsaki, Werbos)

Traditional AI: Puts the emphasis of symbolic representations and thus can not grasp the

essence of intelligence. Critique by: A Clark, H Dreyfus, J. Hawkins,…

Dynamics for Intelligence SDH: Solve the paradox of symbolic nature of intelligence and embodiment Symbols are embedded in the dynamics, they emerge and disappear according to

the context Intentional neurodynamics gives a tool to model intermittent emergent structures


2. State variables: Microscopic pulses and waves; Macroscopic pulse and wave densities.


Synapses convert pulses to waves ��Trigger zones convert waves to pulses


Static sigmoid nonlinearity in wave-pulse conversion and its derivative, dp/dv (amplitude-dependent gain)


3. Multichannel ECoG, Power-law temporal spectrum

A. Multichannel recording from a high density array fixed over the auditory cortex of a rabbit at rest. B. The power spectral density was computed for all 64 ECoGs and averaged. The power-law trend lines (1/f0 and 1/f2.6) were drawn by hand to emphasize the multiple peaks of power in the theta and beta-gamma ranges. From Figs. A1.01 and A1.02 in Freeman (2006)


3. Measures of Gaussianity Amplitude histograms after filtering


Walter J Freeman

University of California at Berkeley

15

Impulse responses of interacting excitatory neurons 4. Linearity of Response Charateristics


The “open loop” evoked potential is derived by single shock electrical stimulation of the olfactory tract in animals under deep anesthesia, which suppresses “spontaneous” electrical activity in bulb and cortex.

KO

KI

KII

K- is for Aharon Katchalsky 4. Linear ODE Description – Open loop (KO)


5. Proof of Stability KIe impulse response; canonical PSDT

A, B. Triangles show the post stimulus time histogram (PSTH) of a representative neuron in response to a weak electric shock, fitted with the solution to a 4th order linear differential equation. The prolonged discharge without inhibitory overshoot is due to mutual excitation. (Freeman, 1975) C, D. The decay rate determines the inflection frequency of the power-law PSDT, and the rise rate determines the exponent, a. The predicted and observed range of the trend lines is 2 < a < 4 in 1/f a. Freeman and Zhai (2009)


6. Hilbert Analysis for Oscillations in EEG


Example of Electrode Array Arrangement on Rabbits Cortex

!

Data observed by olfactory/auditory/visual/somatosensory experiments; Barrie, Freeman (1996).

8x8 arrays of electrodes (size 8mmx8mm) Oscillatory patterns over various sensory areas in visual area


Illustration of Hilbert Analysis

A(t) = [ Re(t)2 + Im(t)2 ] 0.5 P(t) = atan [ Im(t) / Re(t) ];

Phase slips


Coordinated Analytic Phase Differences (CAPD)

Time is on left abscissa, channel order is on right


Transient Dynamics of Brain Operation

Interpretation using the concept of cognitive phase transitions •  Brains are open

thermodynamic systems converting random input with noise into meaning and intelligence in the cognitive cycle of knowledge creation.


II Large-Scale Networks and Graphs

Graph theory basics

•  Transient behavior at various spatial and temporal scales •  Erdos-Renyi, Strogatz-Watts, Albert-Barabasi models •  Scale-free structure and dynamics, black swans and dragon kings •  Percolation and characterization of phase transitions


Transient Events in Nature - Examples


Common Features of Transient Data at Different Spatial and Temporal Scales

Edge of stability

Noise

Criticality

BRAIN EEG

COSMOS G-Doradus

MICROWORLDS Calabi-Yau space


Supernova Explosions

Supernova: Explosion by the end of the life cycle of massive stars Shock Break Out: Event that occurs instants after explosion of a supernova.

Figure: nasa.gov


Fundamental Question: Are Transients Predictable?

Answer A: Not really!

Crisis events are

manifestations of events occurring continuously at all scales. Following scale -free statistics they can happen at

any time, like

Black Swan

Answer B: Yes!

Crisis events may be predictable based on

precursors. They form a new class of phenomena (not scale

free). Sornette (2012) introduced the term

Dragon King


Nuclear Disasters – Statistical Analysis Largest 15 disasters (M$)

Outliers of scale-free behavior

Wheatley et al, 2015)


Probability of Large Terrorist Events

Clauset, Woodard (ann. appl. stat. 2013)


Graph Theory Models of Criticality and Phase Transitions


What ls the Language of the Brains? Not Mathematics !

…”Brains do in “few short steps” what computers do with exquisite numerical precision over many logical steps, because “brains lack the arithmetic and logical depth that characterize computers.”

…“Whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics.”

(J. Von Neumann, 1958.The Computer and the brains, pp. 80-81).


Information versus Knowledge

Information Defined in absolute units Stripped off its content - disembodied Shannon proposed it for information

transmission in engineering systems He never intended using it for biological

systems when the context is crucial Knowledge

Relates to meaning of the sensing What is relevant to the (human) animal Not absolute, but context-dependent

Claude Shannon’s electromechanicalMouse Theseus that can ‘learn’ basedOn ‘teaching’ conditions(C. Shannon, 1917 - 2001)


Historical Sketch on Computing, Mathematics, and Intelligence

Dreams of imitating human behavior (17-18 c.) Toys, puppets Speaking machines, chess machine F. Kempelen -> ‘The Turk’ chess machine

Mechanical computing (18-19 c.) Charles Babbage (1792-1871) Music, textile manufacturing Astronomy calculations Specialized steps:

problem analysis, program design, arithmetics

Electrical computing (1940+)

(C. Babbage)


Computing Machines Principles of modern digital computing

Norbert Wiener Numerical arithmetics Fully electrical machine (no mechanical

components) Binary representation No human interference, rather use built-in

decisions Memory for information storage & recall

First computers ENIAC - Electronic Numeric Integrator &

Calculator, Princeton; JONIAC - Designed by J Von Neumann using programs

Neumann’s Post humus work: “The computer and the brains.” (1958) Warns of the limits of digital computers.

Store in memoryCommand lines‘Von NeumannArchitecture’ (1948)


Universal Computing Paradigm Dominant principle in the 20th century

Disembodied computers

Formulate for any problem in general terms Universal digital machines Departure from biological analogies Very powerful but limited (brains have advantages) Brain computing is signal- and context-dependent

Information theory (Shannon, 1948)

A parallel development supporting disembodiment Turing test (1937)

Decide intelligence Language of the brain? à Not mathematics !!


Success of Traditional AI … Dominant from 50’s to 80’s It has produced significant results

but couldn’t fulfill its promises to create intelligent machines that parallel human intelligence -> rigidity

Chess Again… Deep Blue (IBM) beat Garry Kasparov, 1997 Does this mean Deep Blue is intelligent ? Brute force with massive computing power, Is that enough?


Traditions of Analytical Approach & Modern Science

Scientific approach •  Break the system into small parts with sharp

boundaries •  Understand the behavior of the parts •  Overall behavior is given as sum of the parts

Root of Calculus •  from 17th c., Newton, Leibnitz,… •  Equations of motion, limitations •  Quasi-linear, no analytical solutions

Nonlinear/complex systems: •  New discipline •  Dynamical systems, chaos •  Continuous and discrete models


Status of Analytical Approach to Complex System

Troubles arose The system may contain too many parts Dividing into parts sometimes essential links

are diminished Biological systems are more than sum of parts

Principle of incompatibility (Zadeh ‘73) “…as the complexity of the system increases,

our ability to make precise and yet significant statement on its behaviour diminishes until a threshold beyond which precision and significance are exclusive characteristics”


Philosophical Aspects of Symbolism

Situated intelligence approach (H. Dreyfus, R. Brooks): Prominent example of a philosophical alternative to

symbolism. Following Heidegger and Merleau-Ponty: intelligence

is defined by Dreyfus in the context of the environment. Thus, a preset and fixed symbol system can not grasp the essence of intelligence.

Pragmatic implementations of situated intelligence find their successful applications in embodied intelligence and robotics.

Symbolic Representations Physical Symbol System Hypothesis (Simon, Minsky, Newell): Symbolic systems are necessary and sufficient for intelligent behavior.


Alfred Renyi (1921-1970)

Paul Erdos (1913-1996)

New Mathematics of Networks

Hubs: in www yeast (Barabasi et al, Science, 1999)

Random Graph Phase Transitions (Renyi, Erdos) On the evolution of random graphs (1960)


Phase Transitions

Terminology of phase transition •  Disciplinary dominance: Phase transitions are

defined traditionally in stat physics, mathematics. •  How to transfer this concept to biology/cognitive

domain?

Substantial aspects & Questions •  Are discontinuities present in brain dynamics? If

there are: •  Are they important/substantial to describe the

system’s (brain) functions? •  If they are important: do we have the tools to

describe them?

Answer 1: Yes: Phase Transitions in Graphs/Percolation Answer 2: Yes: Bose-Einstein Condensates/Quantum FT


III Graph Theory for Brains and

Cognitive Models


Modeling Phase Transitions in Brains

Two ways of describing phase transitions between aperiodic basal state and constrained attractor wings

Continuous Neural Dynamical Systems Biologically-inspired K models (Katchalsky): Lumped system of ordinary differential equations ODEs Sensory input induced phase transitions

Discrete Approach to Neural Phase Transitions Transitions controlled by coupling strength E.g.: coupled map lattice CML (Kaneko, 1990; Ishii 1995) Phase transitions in random graphs (Renyi, Erdos, Bollobas,…)

Neuropercolation: generalized random percolation for the interpretation of brain experiments


w  A key difference from conventional NN models: w  Conventional NN: activations typically converge to fixed point w  K models: activations show nonconvergent/chaotic state

w  Hierarchy of K models of increasing complexity: space-time scale w  From K0 to KIV sets

Hierarchical Model of Neural Populations: Freeman K Sets

K0: a single inhibitory KOi or excitatory KOe node second order non-linear transfer function. KI: coupling either two KOe or two KOi nodes population of recurrent K0 units KII: two KOe nodes and two KOi nodes. generate non-zero fixed point or limit cycle behavior. KIII: three layers of KII populations aperiodic oscillations simulating dynamics of cortical regions. KIV: three connected KIII models combine spatial-temporal oscillations at hemisphere level.

Cell à Column à Bulb à Cortex à Hemisphere


Principles of Neurodynamics (Freeman, 1975, 2000)

1. Non-zero point attractor generated by state transition of an excitatory population starting from point attractor with zero activity. This is the function of the KI set.

2. Damped oscillation through negative feedback between excitatory and inhibitory neural populations. Controls the beta-gamma carrier frequency range and it is achieved by inhibitory feedback. KII having low feedback gain.

3. Transition from a point attractor to a limit cycle attractor regulates steady state oscillation of a mixed E-I KII cortical population. KII with high enough feedback gain.

4. Broad-spectral, aperiodic/chaotic oscillations as background activity by combined negative and positive feedback among several KII populations. Achieved by KIII sets.

5. Distributed wave of chaotic dendritic activity that carries a spatial oscillatory patterns. Amplitude modulation AM in KIII.

6. Increase in nonlinear feedback gain that is driven by input to a mixed population, results in the destabilization of background activity and leads to first step of perception. Emergence of AM pattern in KIII.

7. The embodiment of meaning in AM patterns of neural activity shaped by synaptic interactions that have been modified through learning. Learning in KIII layers.

8. Enhancement of macroscopic AM patterns Attenuation of microscopic sensory-driven noise and carrying meaning by divergent-convergent projections. Cortical projections in KIV.

9. Gestalt formation through the convergence of external and internal sensory signals leading to activation of the attractor landscapes. Intentional action-preception cycle in KIV.

10. Global integration of frames at the theta rates through neocortical phase transitions representing high-level cognition Cognitive activity in the KV model.


K-sets Toolbox


Cellular automata and percolation on lattices •  Initial random active sites (0/1)

ü  With probability p

•  Update rule for activation ü  k out of N neighbors active ü  E.g., N=5, k=3 (majority rule) ü  Active remains active

•  Question ü  Is there an all-active final state? ü  == PERCOLATION


Phase Transitions in Random Cellular Automata RCA

Very difficult mathematical problem: Mixture of object with algebraic structure

(CA) and object without topologic structure (random graph).

Solid math results in specific cases Mean field model (all connections are

random): exponential waiting time for switch between metastable states (Balister, Bollobas, Kozma, 2006, RSA)

Local lattice model (all connections local): proof of existence of phase transition with very small random component (Walters et al, 10)

Practical progress through large-scale simulations

Generalization of Ising models, statistical physics tools

l  Neuropercolation Phase Transition: §  Size of connected components of active

sites? §  Number of components ? §  Transition times, waiting times?

l  Phase transition proof steps: §  Small p (p ~ 0) §  Exponential waiting time §  Polynomial transition time §  Transition through finding double-band

configuration

l  Non-locality controls time dimension §  Contraction of exponential waiting time


Neuropercolation (since 2001) - Generalized Percolation -


Neuropercolation Motivation

Structure Function/Dynamics

Percolation: - Random initial à propagate - Cellular automata rule: Arousal, Depression Characterize asymptotics

Neuropil: Densely connected filamentous texture of neural tissue

Activity of neuropil -Subthreshold regime - Possible phase transitions from connectivity/gain change

NxN Lattice/Torus Basic steady-state Oscillations induced by: - Process noise - (Rare) non-local link - 2 layers: excitatory- inhibitory

Erdos, Renyi, 1960; Bollobas, Riordan, 2002.


Critical Spatio-Temporal Clustering in Local Random Cellular Automata

Illustration of model simulations: Activity patterns in lattices in the sub- and supercritical regimes, as well as in the window of phase transition. Notice the emergent giant cluster near the critical probability (pc) similarly as it has been described in the evolution of random graphs. Lattice size is N x N = 128 x 128. p = 0.080 p = 0.120 p= 0.133 p = 0.134 p = 0.136 p = 0.140 Subcritical regime Phase transition Supercritical regime weak- /no clustering prominent clustering clustering / decreases

Mea

n ac

tivity

Time / Iteration step (total number of iterations is 5x106)


Finite-size Scaling in Neuropercolation - Binder Method of 4th order moments -


Chaos in Coupled Neuropercolation Motivated by KIII Freeman Sets

Multi-layer structure Interconnected narrow-band oscillators Biologically motivated by sensory system Produces broad band noise and chaos

Puljic, Kozma, 2010

Progress with HW Modeling and NP Simulations

Research on Chip and NP •  Focus on understanding the behavior by determining

characteristic temporal and spatial scales and transfer characteristics based on data readouts from the existing devices by multiple electrodes;

•  Conduct computational models by KI-level NP using the determined features; i.e., scale-free power spectra and residence time distribution.

Neuropercolation simulations •  Progress with HPC account •  Determined boundaries of large-scale synchronization

domains with narrow-band oscillations •  Developed methods to evaluate proximity of the NP to

criticality in accord with the Binder criterion (correlation length approaches max system size)

•  Ongoing: use of analytic power as a measure of the rate of energy dissipation in the subcritical, critical, and supercritical domains.

Critical cluster size distribution Various ratios of rewiring (0, 25%, 100%)

Synchronized critical oscillations Dash: individual, solid: mean field average

Am

plitu

de

PSD

Application of transient brain

dynamics for Brain-Computer-Interface designs

Neuropercolation: Metastable Amplitude Patterns

Super-Turing Computing

In the Style of Brains

(Kozma, Puljic, Theor. Comp. Sci., 2016)

Hardware Embodiments

Digital domain: VLSI chip •  Neuropercolation model in digital

domain •  24x24 unit array and expandable in

future

Analog Atomic Switch Network •  Gimzewski, UCLA, Nano lab •  Neuropercolation KI single layer

implementation on atomic switches

Neuropercolation Hierarchy •  Interconnected layers can implement

KII, and KIII sets with WJ Freeman •  Demonstrating learning and

classification

Stieg et al, 2012.

Kozma et al, 2013.

Narayan et al, 2012.

Conclusion: Brain Computing

How Brains Create the Cortical Space? •  Evolving cortical tissue from birth to maturity •  This is the neural substrate of transient dynamics

Pioneer cells (t = 0)

Mostly local connections but some non-local (long -range) effects create the topology of the brain space

IV Transient Space-Time Dynamics

in Large-Scale Brain Data The “AHA” Moment

Book on Neural Fields (2016)

Collective dynamics in football stadium Emergent self-organized dynamics with many

interacting components Contributions by: Werbos, Baars, Vitiello,…

Cortical Phase Transitions in ECoG

Freeman (Clin. Neurophysiol.2004)

Sensor array -  64 channels -  Intracranial probes -  Human ECoG -  Rabbit conditioned

reinforcement learning Spatio-temporal Patterns -  Periods of synchrony -  Duration is 150-200ms -  Punctuated by brief

desynchronization~10 ms

Cinematic Theory of Cognition

VIDEO: Type2 Gamma VIDEO: Type3 SA 3D

Phase Cones in Rabbit EEG

Amplitude modulation AM l Metastable AM pattern for

100-200ms l  Embodies meaning

Phase Modulation PM l Numeric fit of dominant

phase cone l Describes rapid

propagations along the cortical surface during transitions

VIDEO: Cones Detection

Vortices in Analytic Amplitude

Intracranial 8x8 EEG array for chronically implanted rabbits at Freeman lab UCB Conditionally stable AA structures for several 100 ms Short term phase cones and vertices for<0.1s Kozma, Freeman, Rodriguez, 2008.

VIDEO: F15X122

°

°

A Carnot-like cycle for perception transforms energy into knowledge.

Transient Dynamics of Brain Operation

Interpretation using the concept of Generalized Carnot Cycle: •  Brains are open

thermodynamic systems converting random input with noise into meaning and intelligence in the cognitive cycle of knowledge creation.

°

°

°

°

increasing firing rates

°

°

increasing firing coherence


°

°

A Carnot-like cycle for perception transforms energy into knowledge.

°

°

°

°


decreasing firing rates


waste heat and entropy

°

° increasing firing coherence

decreasing firing rates


decreasing firing coherence

Tentative Conclusions – On Brain Networks

•  The methodology that has been presented here has successfully allowed us to display large quantities of data for the study and understanding of the neural correlates of higher cognitive functions.

•  Opens up the opportunity of building a data bank of brain dynamics movies, which can be of great support in the learning and understanding the way the brain creates knowledge and meaning.

•  This methodology and undertaking still has much room for

improvement and new ideas, as well as for a broad spectrum of applications for normal and abnormal brain/cognitive conditions.

V Engineering Applications of Brain Graph Models

-  Intelligent/autonomous agents

-  Brain Computer Interfaces

Intentional versus Reactive Agents Intelligent behavior is characterized by the flexible and creative pursuit of endogenously defined goals. Humans and animals are not passive receivers of perceptual information. They actively search for sensory input. To do so intentional systems must by Freeman: ①  PREDICT:

They must form hypotheses about expected future states, and express these as goals such as safety, fuel, or temperature control.

②  TEST BY ACTION: They must formulate a plan of action, and they must inform their sensory and perceptual apparatus about the expected future input in re-afference.

③  SENSE: They must manipulate their sense organs, take information in the form of samples from their sensory ports.

④  PERCEIVE: They generalize, abstract, categorize, and combine into multisensory percepts (Gestalts).

⑤  ASSIMILATE & UPDATE: They use these new data to verify or negate the hypotheses and update the brain state.

A Word of Caution on Symbolism

Jeff Hawkins: On Intelligence •  From Palm Pilot to Brains •  Strong critique of traditional AI •  Suggestion for Intelligence =

memory + prediction using invariant representations in the neocortex

•  Grandmother cells: some cells respond to given image

•  Eg, Jennifer Alliston or Halle Berry cells (Quiroga, Koch, …)

•  Distributed-localized paradox •  Redwood

Dreyfus on Embodiment Using Freeman Neurodynamics Example

Intentional Robotic Testbed SRR-2K NASA Mars Rover Robot

SRR2K onboard PC104+

FSM in C (1Hz) /ide0/

SODAS on DESKTOP

mget Sensory Data

MATLAB runs continuously

mput Control Action

FTP

Sen_cor Sen_imu Sen_hip Sensoryt

Cmd55 [t, φ, d]

Action Selection Stereotypic: Associative: à Contact KIV learning based à Backtrack Reinforced/frustration à Turn away Integrate IMU&Hazcam

Forward/Backward & Angle to turn

Challenges to Noninvasive BCI Techniques High-frequency EEG component is often considered noise

÷  Frequently eliminated by averaging and band-pass filtering.

÷  Recent studies show that this scalp EEG “noise” has cognitively relevant meaning.

Design principles for EEG arrays: ÷  High temporal and spatial frequency!!! ÷  Reduce EMG using biofeedback ÷  Detect sequence of metastable AM patterns

and rapid phase gradients ÷  Measure canonical power spectra ÷  Identify frequency region of self-similarity

and correlate with cognitive behavior

BCI - MINDO Electrode Designs

Tests at Brain Research Center, CT Lin Lab National Chiao Tung University, Taiwan

High-density scalp EEG array (Liao et al, Proc. IEEE, 2012).

Temporal Power Spectrum Density (PSDt)

green Average PSDt of participant two (2) with closed eyes red Average PSDt of participant two (2) with open eyes minimizing artefacts pink Average PSDt of participant two (2) with open eyes and induced blinking artefacts

PSDt for Central 16 Channels Closed Eyes Condition

Participant 1 Participant 2

Log Hz

Log

Pow

er

Signal’s PSDt for the 16 middle channels of EEG array placed on participant’s forehead.

Channels 1 to 16 are represented in a 4x4 array of figures from top left to bottom right.

Log

Pow

er

Log Hz

Log

Pow

er

Log Hz

Blue: PSDt(j), where j =1,…,N over the 16 central channels, N=16 Red: PSDt averaged over the central 16 channels in the array.

Participant 1 Participant 2

Open Eyes Condition

Log Hz X (axis)

Log

Pow

er

Y

(axi

s)

Log Hz

Log

Pow

er

Log Hz

Log

Pow

er

Log Hz

A difference was observed in the average slope (α) of the PSDt between the two modalities (open eyes and closed eyes) and it is more prominent with Participant 2.

PSDt slope (α) averaged over the 48 channels: Participant 1: Closed eyes α=-1.61; Open eyes α=-1.52 Participant 2: Closed eyes α=-1.82; Open eyes α=-1.44

Spatial Power Spectrum Density (PSDx)

green Average PSDx of participant 2 with closed eyes red Average PSDx of participant 2 with open eyes minimizing artefacts pink Average PSDx of participant 2 with open eyes & induced blinking artefacts

Open Eyes with blinking Open Eyes without blinking

Observations on Spatial PSDx

- Marked difference in the slope of PSDx between closed eye and open eye modalities - In open eye condition, when looking at light source, the slope gets closer to 0 Blue: PSDx(i), i=1,…,T temporal window index Red: PSDx averaged over the temporal windows

Closed Eyes

DATA$

INTERPRETATION$

PHASE$SPACE$

Integrated BCI Approach with MINDO64

COMPARE((

HIGH,DENSITY(EXPERIMENT((ECOG/fMRI/MEG/EEG)((

RANDOM(GRAPH(MODEL(,(NEUROPERCOLATION(

!!!!Mul%channel ! !Time/frequency ! ! !!Transient!!!!!!!Brain!Monitoring ! !!!!Domain!Data ! !!!Desynchroniza%on!

!!!!Mul%layer ! !!!!!!Simulated!Transient ! !!!Phase!Transi%ons!!Neuropercola%on ! !!!!!Time!series ! ! !!!!!near!Cri%cality!!!!!!!!!!!!!!

VALIDATION((((!

!Metrics!(PSD!slope,!freq.,!…)!

MODEL(ADAPTATION,(TUNE(TO(CRITICALITY(Control!parameters!(Inhibi%on,!Noise,!Rewiring),!Learning!!

(a)!

(b)! (c)!

(d)!

Conclusions on Mindo BCI

1.  We have successfully trained participants through biofeedback to reduce EMG artifacts during different experimental conditions. Obtained acceptable signal/noise ratio with a relatively small amplitude 60 Hz component.

2.  We can discriminate between experimental modalities: open eyes

and closed eyes, with and without visual stimulus, using both PSDt and PSDx. The spatial analysis based on PSDx adds an immense value to the overall understanding of brain dynamics.

3.  For both open and closed eyes, the PSDt and PSDx has 1/fa

behavior, where the power exponent is 1< a < 2 (pink noise) •  There is a clear difference in the slope a between closed and

open eye conditions. •  In the presence of artificially induced EMG artifacts, the PSDt

still shows 1/fa behavior, where the exponent is 2 < a < 3 (black noise).

General Conclusions on BCI

1.  BCI technology is available based on noninvasive scalp electroencephalogram (EEG) with high temporal and spatial resolution.

2. Monitoring intermittent synchronization-desynchronization

effects and carry cognitive content as neural correlates of cognitive functions.

3. Hardware implementations can be applied for real-time

monitoring and modeling healthy individuals as well as patients with diseases, e.g., epilepsy, schizophrenia, dementia. chip.

The information extracted from high-density EEG array manifests neural correlates of higher cognitive behaviors.

Transient Stars Modeled

by Neuropercolation

Transients in Brains and Stars

Case Study: Gamma Doradus Stars

•  Doradus stars in Large Magellan cloud

•  With magnitude variations due to non-radial gravitational modes (g-modes)

•  They have periods between 0.3 and 3 days.

•  Between 1 and 5 periods can be found.

/ Catalina Elzo Vera / Pablo Estévez Valencia /

97

Interconnection between oscillators

98

Connections between

oscillators

Parameters Optimized By PSO Algorithm: -  Internal Dimension -  Non-locality Ratio -  Inhibition Ratio -  External Geometry -  Number of Nodes

Options of inter-node connectivity

99 / Catalina Elzo Vera / Pablo Estévez Valencia /

Connectivity Structure – Options

100

•  Inspired by the modes of oscillation

•  “Spherical” connections

•  As many “layers” as we want

/

Iterations of Lightcurve Modeling of KIC 002575161

101

Delayed Embedding of KIC 9240041

102

Iterations: Gamma Doradus KIC 9240041

103

Modeling Error & Frequency Spectrum

104 /

KIC 9240041

Modeling KIC 10224094 (quasi-chaos)

105 /

Conclusions: Astroinformatics

106

•  Neuropercolation modeling shows promise besides biomedicine, including star curves modeling and interpretation.

•  Efficient encoding schemes, e.g., with translational, rotational symmetry considerations are being implemented.

•  Optimization of model parameters using PSO techniques.

•  Future work: adaptation to automatic processing of massive data (LSST and other telescopes)

Looking Ahead

Scientific Revolutions and Paradigm Shifts, T. Kuhn, 1970’s

graph theory approaches to neural structures and dynamics

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