Circus, Circuits
Interesting Neural Networks:Some actually occur in brains; some are
hypotheses
Owl Audition• 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.
B
C
D
A
E
From Left Ear
From Right Ear
Far Right
Far Left
Center-Surround Cells
C
C C
C
S
S
S
S
S
S
S
S
SS
Visual Field
S
C Center
Surround
Inhibit
CS
CS firing pattern
Stim
ulu s
“ON center OFF surround” cell
Retina
Strong
Medium
Weak
Ganglia
Brain
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)
Receptors
Bipolar Cells
Retinal Ganglion
Light
Receptors
Light
CC C
C
S
S
S
S
SS
S
SSS
On-Center (Off-Surround) Off-Center (On-Surround)
These are non-intersectingpathways but are drawntogether to illustrate theirsimilarities.
ExciteInhibit
Line Detectors
On-Centers
Off-Centers
45o Line
Retinal Ganglia
Visual Cortex
To higher levelsof the visual cortex
Motion DetectorsRiechard Detector (1961) - based on the fly’s visual system
Delay
Delay
t1 t2 t3+
Delay
450 bar movingleft to right
t2+
t3+2
• Works best when delay = t2 - t1 = t3 - t2• = normal (non-delayed) transmission time
Coincidence detectors =>only fire when all inputsare ON simultaneously.
t3
Lateral Inhibition Lines
• Neurons that stimulate themselves and inhibit their near neighbors function as filters
1 2 3 4 5
Firi n
g R
ate
Neuron
Firin
g R
ate
Input
Output
ExciteInhibit
Lateral Inhibition in Visual Pathways• Grossberg, S. (2003) in The Handbook of Brain Theory and Neural
Networks, pp. 594-600.
Retinal ON Cell
LGN
V1(6)
V1(2/3)
V1(4)
V2(6)
V2(4)
V2(2/3)
ExciteInhibitInterneuron• 6 - 4 - 2/3 pathway/loop is self-excitatory
• Similar lateral inhib topology in V1 & V2
Central Pattern Generators (CPGs)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
Overhead view ofhorse, goat, dog??
t2
t3t1
t4
Walking gait: First move left rear leg, then left front, then right rear, then right front.
1/4
1/20
3/4
Standard Notation:Fractions = Phase diffs
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.
LeftFront
RightRear
RightFront
LeftRear
AR1
AL2
AL1
AR2
Inter-loop delay
Intra-loop delay
By adjusting theseTWO delay times,we can generateall standard gait patterns forN-legged animals!!
1/4
1/2
3/4
0
0
3/4
1/2
1/4
1/41/2
Walking
1/4
0
1/4
0
1/2
3/4
1/2
3/4
1/40
Jumping
1/20
1/2
0
0
1/2
01/2
Pacing
1/2
1/2
0
0
1/2
1/2
0
0
1/21/2
Trotting
0 1/2
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
B
D
C
t1 t2 t3
Time Signatures
A B C D
t1 0 1 0 1
t2 1 1 1 0
t3 1 1 0 0
t4 0 0 1 1
t4
Timing Circuit
A B
C
D
Cerebral Cortex
Neural Oscillators from 10-40 Hz
S
StriatumSNc
SNr
STN
DopamineSignal => Learn!Excite
Inhibit
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
Learning a Time Signature
• 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.
B C
D
A
S
B C
D
A
S
Low
High Low
High
STOP!!LEARN!!
Cricket Phonotaxis• Webb, B. (2001). Biorobotics: Methods & Applications, Ch. 1.• Female Crickets only respond to songs with particular carrier
frequencies and syllable durations.
LeftEar Drum
RightEar Drum
• Syllable Duration• Carrying Frequency = 1/Inter-syllable period
BugOff!
Preferred Carrier FrequencyDistance 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.
Peak
Trough
Time T
Time T+P/2
• 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!!
Eardrums R L
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.
ANR
RightEar
LeftEar
ANL
MNR MNL
AuditoryNeurons
MotorNeurons
Turn Left
Turn Right
• 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
Leaky Integrate-and-Fire Neural Models
tmdVi/dt = b(EL - Vi) + awijzj
zj = (1 + eVi)-1 {Standard sigmoidal transfer function}
Vi = voltage inside the neuronEL = 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
Vi
EL
z1*wi1
z2*wi2
z3*wi3
zi
Leak
Leak Integrate
AP = Voltage Spike• Although the voltage of a neuron changes constantly, only large abrupt
changes (action potentials) can be transmitted to other neurons.
0 mV
+40 mV
-65 mVResting Potential
Overshoot
RisingPhase
FallingPhase
Undershoot
Na+ gates open.Na+ enters cell.
K+ gates open.K+ leaves cell.Na+ gates still open
Na+ gates close.K+ gates still open.
K+ gates close.
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
Vj
zj
Vi
S
wij
z j
t
wij
t
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.
• The cricket turns left. It is attracted to the song.
ANL Response
Incoming soundSyllable
• 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.
MNL Response
Null Poeng• Stimulus again from left side, but now the syllables are very short and frequent..
• The cricket is not interested.
ANL Response
Incoming soundSyllable
• Neuron ANL integrates the inputs from the left ear drum and fires constantly, with very few significant gaps.
MNL Response
• 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.
Another Loser• Stimulus again from left side, but now the syllables are very long, with a large gap between syllables..
• This is cricket is very picky!
ANL Response
Incoming soundSyllable
• Neuron ANL integrates the inputs from the left ear drum and fires long sets of pulses with long gaps.
MNL 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.