1. Neural Algebra and Consciousness Erwin Engeler ETH Presented by Bruno Buchberger, RISC, at “Algebraic Biology 2008”, August 1, 2008

2. I am not giving Engeler’s talk. I only serve as a pointer. I give a personal remark on the paper at the end.

3. Objective Neural Algebras Consciousness Meditation

4. Objective A mathematical model of “thought”, “perception”, “memory”, “motor activation” …

5. Neural Algebras Basic objects M, N, …: “firing patterns of neurons”. One level of abstraction higher than neural networks. Composition of firing patterns: R = M. N Functional / structural problems (e.g. the existence of “consciousness”): Algebraic solution of equations in neural algebras.

6. Definition of the notion of “firing pattern” in a neural network: technically involved (“nested substitution patterns”) Knowledge base of findings in neurology of humans and animals can / could be cast in terms of neural algebras. Mathematical properties (theorems) of neural algebras: –2.1 All fixpoint equations have a solution –2.2 Certain neural algebras are “combinatory algebras” –2.3 All computable relations are representable in sufficiently rich neural algebras

7. Consciousness Consciouness: The power of self-reflection “The power of observing itself as being conscious and as consiously planning and acting”: C. ( C U U M i. C) = C i This fixpoint equation can be solved in sufficiently rich neural algebras (“primitive” animals sufficient; “colonies” of individuals also a possible implementation).

8. Some mathematical findings about “consciousness”: –3.1 C always has a base in one or more cycles –3.2 C can be expanded along any outgoing edge –3.3 C never expands into cycle free “stimulus and and response” structure.

9. “Meditation” The “Western” approach to “understanding”, e.g. understanding consciouness: Intellect (“mathematics”), i.e. think deeply until you have a “model” of the phenomenon. The Eastern approach to “understanding”, e.g. understanding consciousness: Experience (“be”) consciousness: don’t live in the model, allow thinking to rest in silence, … (“Meditation”)