Derek Harter University of Memphis May 17, 2004 Aperiodic Dynamics and the Self-Organization of Cognitive Maps in Autonomous Agents Derek Harter University of Memphis May 17, 2004 FLAIRS 2004

Discrete difference eq. ANN KA K Discrete difference eq. Continuous Feed-Forward Recurrent Back-Propogation Hebbian

Table 1: Characterization of the hierarchy of K-sets KA – Set Hierarchy Table 1: Characterization of the hierarchy of K-sets Type Structure Inhrt Dynamics Exs. in brain* KA-0 Single Unit Nonlinear I/O function All higher level K sets are composed of K0 units KA-I Populations of excitatory or inhibitory units Fixed point convergence to zero or nonzero value PG, DG, BG, BS KA-II Interacting populations of excitatory and inhibitory units Periodic, limit cycle oscillations; frequency in the gamma band OB, AON, PC, CA1, CA3, CA2, HT, BG, BS, Amygdala KA-III Several interacting KII and KI sets Aperiodic, chaotic oscillations Cortex, Hippocamp, Midline Forebrain KA-IV Interacting KIII sets Spatio-temporal dynamics with global phase transitions (itinerancy) Hemisphere cooperation of cortical, HF and MF by the Amygdala * Notations: PG – periglomerular; OB - olfactory bulb; AON - anterior olfactory nucleus; PC- prepyriform cortex; HF - hippocampal formation; DG - dentate gyrus; CA1, CA2, CA3 - curnu ammonis sections of the hippocampus; MF - midline forebrain; BG - basal ganglia; HT - hypothalamus; DB - diagonal band; SP - septum KA-0 Unit Difference Equation: Transfer function: KA-III KA-Ie/i E1 E2 I1 I2 Layer 1 d receptors Layer 2 Layer 3 E1 E2 Named in honor of Katchalsky, a famous and influential neuroscientist) Maximal slope of transfer function is displaced to the excitatory side of the resting level. An excitatory input to the population raises the activity level of its neurons, and also increases their sensitivity to input from each other, so that they interact more strongly. The asymmetry of the sigmoid function means that the OB is destabilized by input. I1 I2 KA-II E1 E2 I1 I2 + -

KA-III =0.058 =0.-044 =0.0109 =0.152 E1 E2 I1 I2 Layer 1 d receptors Layer 2 Layer 3

KA, K-Set and Rat Power Spectra KA-III model – Layer 1

Evolution of Intentional & Deliberative Behavior in Complete Biological Organisms 3.5 billion years: single-cell entities 550 million years: fish & vertebrates 430 million years: insects 370 million years: reptiles 330 million years: dinosaurs 250 million years: mammals 120 million years: primates 18 million years: great apes 2.5 million years: man 5000 years: writing Basic Limbic System Primitive Hippocampus Long-term memory Beyond stimulus/response Episodic Memory Cognitive Maps Intentional Behavior & Deliberative Actions

Limbic System: Simplest Complete Intentional System Far from equilibrium, therodynamic systems Mechanisms of self-organization competition cooperation autocatalytic loops hierarchy & mesh Aperiodic dynamics Expectation or reafference Embodiment environment/organism coupling Small worlds type divergent-convergent systems connections (Kozma, Freeman & Erdi, 2002)

Architecture of Hippocampal Simulation

Formation of AM Pats in KA-III c d b (Loc 1) (Loc 2) (Loc 3) (Loc 4) (Loc 5) (Loc 6) (Loc 7) (Loc 8) (Test d) (Test c) (Test b) (Test a)

Comparison of Closest AM Pattern Target Loc-Test Closest 1-a 1-b 1-b 1-a 1-c 1-b 1-d 1-a 2-a 2-d 2-b 2-c 2-c 2-b 2-d 2-b 3-a 3-b 3-b 3-a 3-c 3-b 3-d 3-a 4-a 4-c 4-b 4-d 4-c 4-a 4-d 4-b 5-a 5-b 5-b 5-a 5-c 5-d 5-d 5-c 6-a 6-b 6-b 6-d 6-c 6-b 6-d 6-b 7-a 7-d 7-b 7-d 7-c 7-d 7-d 7-a 8-a 8-c 8-b 8-d 8-c 8-a 8-d 8-c a c d b b d c a

Conclusion The K/KA neural population model highly recurrent multi-layer biologically grounded model aperiodic neurodynamics The Limbic system model with the primitive hippocampus represents an enormous leap in behavioral complexity in biological organisms Intentional behavior Deliberative behavior KA-III can develop representations of the environment using aperiodic dynamics.