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Circus, Circuits

Circus, Circuits. Interesting Neural Networks: Some actually occur in brains; some are hypotheses. Owl Audition. Far Right.

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Circus, Circuits

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  1. Circus, Circuits Interesting Neural Networks: Some actually occur in brains; some are hypotheses

  2. Owl Audition Far Right • The Barn Owl uses delay lines and coincidence detectors (neurons that only fire when both pre-synaptic axons are simultaneously depolarized) to locate objects in horizontal and vertical plane. From Left Ear A B C D E From Right Ear Far Left

  3. S S S C S S C C C S S S S S Center-Surround Cells “ON center OFF surround” cell CS Visual Field Brain Ganglia CS firing pattern Strong C Center Surround Inhibit Retina Medium S Stimulus Weak

  4. S S S C S S C C C S S S S S On-Center -vs- Off-Center Retinal Ganglion Cells • The primary visual receptors (rods & cones) actually turn OFF when hit by photons (light) and are ON when they detect dark spots (Hubel, Eye, Brain and Vision, 1988, pg. 54) On-Center (Off-Surround) Off-Center (On-Surround) Bipolar Cells Light Light Receptors Receptors These are non-intersecting pathways but are drawn together to illustrate their similarities. Excite Inhibit Retinal Ganglion

  5. Line Detectors Retinal Ganglia Visual Cortex On-Centers 45o Line Off-Centers To higher levels of the visual cortex

  6. Motion Detectors • Works best when delay = t2 - t1 = t3 - t2 •  = normal (non-delayed) transmission time Riechard Detector (1961) - based on the fly’s visual system 450 bar moving left to right t3 t2 t1 t3+ Delay t3+2 Delay Delay t2+ Coincidence detectors => only fire when all inputs are ON simultaneously.

  7. Lateral Inhibition Lines Firing Rate Output • Neurons that stimulate themselves and inhibit their near neighbors function as filters Excite Inhibit 1 2 3 4 5 Firing Rate Input Neuron

  8. Excite Inhibit Interneuron Lateral Inhibition in Visual Pathways V2(2/3) V1(2/3) • Grossberg, S. (2003) in The Handbook of Brain Theory and Neural Networks, pp. 594-600. V2(4) V1(4) V2(6) V1(6) LGN • 6 - 4 - 2/3 pathway/loop is self-excitatory • Similar lateral inhib topology in V1 & V2 Retinal ON Cell

  9. t2 t3 t1 t4 1/4 1/2 0 3/4 Central Pattern Generators (CPGs) Overhead view of horse, goat, dog?? Neural circuits for generating simple, repeated patterns of activity. E.g. gait patterns in N-legged animals. Ian Steward (1998). Life’s other secret. Ch.9 Standard Notation: Fractions = Phase diffs Walking gait: First move left rear leg, then left front, then right rear, then right front.

  10. AR1 AL2 AL1 AR2 Left Front Right Rear Right Front Left Rear Generic Gait Generator • Each animal species can perform many different gaits. • Do we need a different wiring pattern for each gait? • No! (Golubitsky, Stewart, Collins, Buono (1997)) • Goal: A single circuit with adjustable delay times. • Solution: For an N-legged animal, 2 cross-linked N-neuron loops. Inter-loop delay Intra-loop delay By adjusting these TWO delay times, we can generate all standard gait patterns for N-legged animals!!

  11. 1/4 0 1/4 1/2 Walking Jumping 3/4 1/4 3/4 3/4 1/2 0 1/2 1/2 1/4 3/4 1/4 1/4 0 1/2 0 0

  12. 0 0 1/2 1/2 0 1/2 1/2 1/2 Pacing Trotting 0 1/2 1/2 0 0 1/2 0 1/2 0 1/2 0 1/2

  13. Brain Clocks • Wright, Karen,”Times of our Lives”, Scientific American, Sept. 2002 • In the cerebral cortex, a collection of neurons with different firing patterns enables us to record and reuse specific time intervals. A Time Signatures B C D t1 t2 t3 t4

  14. 1. A start signal (e.g. Dance instructor says ”Begin”): STN excites SNr, which then inhibits all cortical oscillators, so they essentially RESET to off. 2. Oscillators then resume their normal diverse firing patterns, from same init state. 3. A stop signal (e.g. Dance instructor…): SNc releases dopamine into striatum, causing striatal cells to record the current time signature via Hebbian Learning Timing Circuit Neural Oscillators from 10-40 Hz Cerebral Cortex B A C D STN S SNr Striatum SNc Dopamine Signal => Learn! Excite Inhibit

  15. Learning a Time Signature Low High C C B High B D D Low A A • Non-associate Learning: Strengthen pre-synaptic axon since: a) it fired/depolarized, and b) significant event (STOP) signalled. • After learning, S will only fire when B & D are active (i.e. after a time interval of duration = t1). Details are unclear as to whether A & C develop inhibitory links to S. • In future (e.g. when repeating the dance), the instructor still says ”Go”, which again resets the cortical oscillators, but now the brain generates its own ”STOP” signal in the striatum, when S fires => student has learned t1! • Given enough diverse oscillators, student can learn ANY interval. STOP!! LEARN!! S S

  16. Left Ear Drum Right Ear Drum Cricket Phonotaxis • Webb, B. (2001). Biorobotics: Methods & Applications, Ch. 1. • Female Crickets only respond to songs with particular carrier frequencies and syllable durations. Bug Off! • Syllable Duration • Carrying Frequency = 1/Inter-syllable period

  17. Preferred Carrier Frequency Eardrums R L Time T Peak Distance between the two ear-drums is the critical determinant. If it’s ONE QUARTER the song’s inter-syllable wavelength, then the eardrums vibrate most strongly. Here P = period of the sound wave. • From T to T+P/4, the peak travels across the body and meets the right eardrum, causing it to vibrate, thus generating a new peak. • From T+P/4 to T+P/2, the new peak travels exactly 1/4 wavelength = ear-to-ear distance. • At time T+P/2, the left ear has a) a trough on the outside, and b) a peak on the inside. • That’s a max pressure difference => the eardrum is maximally stimulated. • The cricket is happy!! Trough Time T+P/2

  18. Preferred Syllable Duration • Appears to be determined in the brain, but details only partially known. • Biorobotics researchers (Webb et. al.) provide minimal ANNs that are sufficient explanations. Turn Right Turn Left • Each auditory neuron stimulates the corresponding motor neuron and inhibits the opposite motor neuron. • Each of the 4 neurons has a very detailed (but standard) model: leaky integrate-and-fire • AN => MN synapses are temporarily depressed after the AN fires Motor Neurons MNR MNL Auditory Neurons ANR ANL Right Ear Left Ear

  19. Leaky Integrate-and-Fire Neural Models Leak Integrate tmdVi/dt = b(EL - Vi) + awijzj zj = (1 + eVi)-1 {Standard sigmoidal transfer function} Vi = voltage inside the neuron EL = voltage outside the neuron (standard value: -55mV) zj = firing rate of neuron j wij = synaptic weight from neuron j to neuron i. a: excitation factor, b: leakage factor, tm = time scaling factor z1*wi1 zi Vi z2*wi2 Leak z3*wi3 EL

  20. AP = Voltage Spike Overshoot • Although the voltage of a neuron changes constantly, only large abrupt changes (action potentials) can be transmitted to other neurons. +40 mV K+ gates open. K+ leaves cell. Na+ gates still open Na+ gates close. K+ gates still open. 0 mV Rising Phase Falling Phase Na+ gates open. Na+ enters cell. K+ gates close. -65 mV Resting Potential Undershoot

  21. Habituation When a neuron fires weakly, but frequently, its axonal synapses weaken. After a little rest, the synapse returns to normal strength. tmdwij/dt = c(wij(*) - wij) - S(zj) wij(*): base value for wij S(zj) = stimulus function; lower zj => higher S wij S t zj Vj Vi wij zj t

  22. Syllable Incoming sound ANL Response MNL Response Preferred Syllable Duration • Assume a stimulus on the left side of the cricket. • High frequency (short wavelength) sound has a quickly-decaying amplitude with distance, so the left ear gets a stronger signal than the right. • Neuron ANL integrates the inputs from the left ear drum and fires groups of pulses with durations = syllable durations. • This inhibits motor neuron MNR but stimulates MNL, which integrates the inputs from ANL and eventually begins to fire. However, it integrates more slowly than ANL and therefore fires less frequently. • The cricket turns left. It is attracted to the song.

  23. ANL Response MNL Response Null Poeng Syllable Incoming sound • Stimulus again from left side, but now the syllables are very short and frequent.. • Neuron ANL integrates the inputs from the left ear drum and fires constantly, with very few significant gaps. • This inhibits motor neuron MNR and stimulates MNL. • But, now the ANL-MNL synapse habituates due to the constant firing of ANL (and hence no break in which to regain strength). • So the signals that ANL sends to MNL are WEAK, and MNL never integrates enough charge to fire. • The cricket is not interested.

  24. MNL Response Another Loser Syllable Incoming sound • Stimulus again from left side, but now the syllables are very long, with a large gap between syllables.. • Neuron ANL integrates the inputs from the left ear drum and fires long sets of pulses with long gaps. ANL Response • This inhibits motor neuron MNR and stimulates MNL. • But, now the gap is too long: MNL almost fires during a syllable, but then a lot of voltage LEAKS out during the inter-syllable gap. • So although ANL’s signals are strong, MNL leaks too much and can never integrate enough charge to fire. • This is cricket is very picky!

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