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.
Far Right
From Left Ear
A
B
C
D
E
From Right Ear
Far Left
Center-Surround Cells
“ON center OFF surround”
cell
S
S
S
C
Visual Field
S
S
C
C
C
S
S
S
S
C
Center
S
CS
Brain
Ganglia
CS firing pattern
Strong
Retina
Medium
S
Surround
Inhibit
Weak
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
S
S
S
Receptors
S
C
C
S
C
C
S
S
Receptors
S
S
S
Excite
Inhibit
Retinal Ganglion
These are non-intersecting
pathways but are drawn
together to illustrate their
similarities.
Line Detectors
Retinal Ganglia
Visual Cortex
On-Centers
45o Line
Off-Centers
To higher levels
of the visual cortex
Motion Detectors
Riechard Detector (1961) - based on the
fly’s visual system
• Works best when delay = t2 - t1 = t3 - t2
•  = normal (non-delayed) transmission time
450 bar moving
left to right
t3
t1
t2
t3+
Delay
Delay
t3+2
Delay
t2+
Coincidence detectors =>
only fire when all inputs
are ON simultaneously.
Lateral Inhibition Lines
Output
Excite
Inhibit
1
2
3
4
5
Input
Neuron
•
Neurons that stimulate themselves and inhibit their near neighbors function as
filters
Lateral Inhibition in Visual Pathways
• Grossberg, S. (2003) in The Handbook of Brain Theory and Neural
Networks, pp. 594-600.
V1(2/3)
V2(2/3)
V1(4)
V2(4)
V1(6)
V2(6)
LGN
Retinal
ON Cell
• 6 - 4 - 2/3 pathway/loop is self-excitatory
• Similar lateral inhib topology in V1 & V2
Excite
Inhibit
Interneuron
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 of
horse, goat, dog??
Standard Notation:
Fractions = Phase diffs
t2
t4
1/4
3/4
t1
t3
0
1/2
Walking gait: First move left rear leg, then left front,
then right rear, then right front.
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.
AL2
AR2
Inter-loop delay
AL1
AR1
Intra-loop delay
Left
Front
Right
Front
Left
Rear
Right
Rear
By adjusting these
TWO delay times,
we can generate
all standard gait
patterns for
N-legged animals!!
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
1/4
1/2
1/4
0
Pacing
0
Trotting
1/2
1/2
0
1/2
0
1/2
0
0
1/2
1/2
0
0
1/2
0
1/2
0
1/2
1/2
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
Time Signatures
B
C
D
t1
t2
t3
t4
A
B
C
D
t1 0
1
0
1
t2 1
1
1
0
t3 1
1
0
0
t4 0
0
1
1
Timing Circuit
Neural Oscillators from 10-40 Hz
Cerebral Cortex
A
B
C
D
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
STN
S
SNr
Striatum
Excite
Inhibit
SNc
Dopamine
Signal
=> Learn!
Learning a Time Signature
High
B
Low
Low
C
High
STOP!!
LEARN!!
•
•
•
C
D
D
A
A
•
B
S
S
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.
Cricket Phonotaxis
• Webb, B. (2001). Biorobotics: Methods & Applications, Ch. 1.
• Female Crickets only respond to songs with particular carrier
frequencies and syllable durations.
Left
Ear Drum
• Syllable Duration
• Carrying Frequency = 1/Inter-syllable period
Right
Ear Drum
Bug
Off!
Preferred Carrier Frequency
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.
Eardrums
R L
Time T
Peak
•
Time T+P/2
Trough
•
•
•
•
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!!
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
Motor
Neurons
MNR
MNL
Auditory
Neurons
ANR
ANL
Right
Ear
Left
Ear
• 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
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
z2*wi2
zi
Vi
Leak
z3*wi3
EL
AP = Voltage Spike
•
Although the voltage of a neuron changes constantly, only large abrupt
changes (action potentials) can be transmitted to other neurons.
Overshoot
+40 mV
K+ gates open.
K+ leaves cell.
Na+ gates still open
Na+ gates close.
K+ gates still open.
0 mV
Na+ gates open.
Na+ enters cell.
Rising
Phase
Falling
Phase
K+ gates close.
-65 mV
Resting Potential
Undershoot
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
Vj
zj
zj
wij
t
Vi
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.
Syllable
Incoming sound
• Neuron ANL integrates the inputs from the left ear drum and fires
groups of pulses with durations = syllable durations.
ANL Response
• 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
• The cricket turns left. It is attracted to the song.
Null Poeng
•
Stimulus again from left side, but now the syllables are very short and frequent..
Syllable
Incoming sound
• Neuron ANL integrates the inputs from the left ear drum and fires
constantly, with very few significant gaps.
ANL 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.
MNL Response
• The cricket is not interested.
Another Loser
•
Stimulus again from left side, but now the syllables are very long, with a large gap
between syllables..
Syllable
Incoming sound
• 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.
MNL Response
• This is cricket is very picky!