work-storage-byTopic-neuroscience-quals-Q4slides

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\emph{...suppose all we had were cells with true Hodgkin-Huxley squid axon dynamics. Is that enough for sensory coding? Computation? Muscular control?}

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\emph{...suppose all we had were cells with true Hodgkin-Huxley squid axon dynamics. Is that enough for sensory coding? Computation? Muscular control?}

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The short answer: yes

Slightly longer answer: yes, I think so


Contents


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\emph{...suppose all we had were cells with true Hodgkin-Huxley squid axon dynamics. Is that enough for sensory coding? Computation? Muscular control?}

For most of the talk I'll focus on the "computation" part.

Answering this question requires defining some terms:

I'll adopt a paradigm popular in computer science, that of the Turing machine


Turing machine paradigm


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Turing machine paradigm cont'd

Computation

Sufficient for computation

If, given any function which some Turing machine can compute, you can compute it too, then, then you can do "computation".


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Turing machine paradigm cont'd


Turing machine paradigm cont'd

$\therefore$

Given a mechanism, we formally show that it is "sufficient for computation" by proving that it can simulate a Universal Turing Machine. Mechanisms which fulfill this criteria are called "Turing-complete".


Turing machine paradigm cont'd

Our question becomes:

\emph{are networks with Hodgkin-Huxley neurons capable of simulating a Universal Turing Machine?}


Turing machine paradigm cont'd

$\exists $ complications, but we don't have time to discuss them


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Turing machine paradigm cont'd

Complications

The requirement that Turing machines have $\infty$ memory complicates things.


To my knowledge, no one has formally proved that H-H neural nets are Turing-complete


Overview of my argument


Many simpler neural net types are Turing-complete


A Spike Response Model neuron


A Spike Response Model neuron


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A Spike Response Model neuron

spiking


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A Spike Response Model neuron cont'd

Synapses

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spatial and temporal summation

\begin{eqnarray*} P(t) = \sum_{i \textrm{ is an incoming synapse}} \sum_{s \textrm{ is a previous spike}} w_i e_i(t - t_s)
\textrm{(where $e_i()$ is the response function of synapse $i$, and $w_i$ is its weight)} \end{eqnarray*}


A Spike Response Model neuron cont'd

Cares about \textbf{one thing}: when spikes occur


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A Spike Response Model neuron cont'd

Networks built out of these neurons are called SNNs.


Spike Response Model neurons can approximate Hodgekin-Huxley dynamics

(Kistler et al '97) attempts to fit SRM model to mimic the Hodgkin-Huxley equations.

SRM predicts the timing of almost \textbf{90\%} of the spikes correctly.


Do SNN computability results apply to Hodgkin-Huxley?


Do SNN computability results apply to Hodgkin-Huxley?


Do SNN computability results apply to Hodgkin-Huxley?

\bigskip

Formally, no; but they strongly suggest that H-H nets are Turing-complete


Sensory and motor functions


Why the zoo?


Why the zoo?


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Examples


Why the zoo? cont'd

$\therefore$ Hodgkin-Huxley probably "sufficient" in Turing paradigm, but creatures built out of only these would probably be either simple or slow (and would need huge brains)


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