Circus, Circuits

1
Circus, Circuits

• Interesting Neural Networks
• Some actually occur in brains some are hypotheses

2
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
3
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
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

• Riechard Detector (1961) - based on the
• flys visual system

• Works best when delay t2 - t1 t3 - t2
• ? normal (non-delayed) transmission time

450 bar moving left to right
t3
t2
t1
t3?
Delay
t32?
Delay
Delay
t2?
Coincidence detectors gt only fire when all
inputs are ON simultaneously.
7
Lateral Inhibition Lines
Firing Rate
Output
Excite Inhibit
1
2
3
4
5
Firing Rate
Input
Neuron

• Neurons that stimulate themselves and inhibit
  their near neighbors function as filters

8
Lateral Inhibition in Visual Pathways

• Grossberg, S. (2003) in The Handbook of Brain
  Theory and Neural Networks, pp. 594-600.

V2(2/3)
V1(2/3)
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
Central Pattern Generators (CPGs)

• Neural circuits for generating simple, repeated
  patterns of activity.
• E.g. gait patterns in N-legged animals.
• Ian Steward (1998). Lifes other secret. Ch.9

Overhead view of horse, goat, dog??
Standard Notation Fractions Phase diffs
Walking gait First move left rear leg, then left
front, then right rear, then right front.
10
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
Walking
Jumping
3/4
1/4
3/4
3/4
0
1/2
1/2
1/2
1/4
3/4
1/4
1/4
1/2
0
0
0
12
Pacing
Trotting
0
1/2
1/2
0
1/2
0
1/2
0
0
1/2
1/2
0
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
A B C D
t1 0 1 0 1
t2 1 1 1 0
t3 1 1 0 0
t4 0 0 1 1
B
C
D
t1
t2
t3
t4
14
Timing Circuit
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
Neural Oscillators from 10-40 Hz
Cerebral Cortex
B
A
C
D
STN
S
SNr
Striatum
SNc
Dopamine Signal gt Learn!
Excite
Inhibit
15
Learning a Time Signature
Low
High
C
C
B
High
B
D
D
Low
A
A
STOP!! LEARN!!
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 gt student has learned t1!
• Given enough diverse oscillators, student can
  learn ANY interval.

16
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

• Distance between the two ear-drums is the
  critical determinant. If its ONE QUARTER the
  songs inter-syllable wavelength, then the
  eardrums vibrate most strongly. Here P period
  of the sound wave.

Eardrums R L
Time T
Peak

• From T to TP/4, the peak travels across the body
  and meets the right eardrum, causing it to
  vibrate, thus generating a new peak.
• From TP/4 to TP/2, the new peak travels exactly
  1/4 wavelength ear-to-ear distance.
• At time TP/2, the left ear has a) a trough on
  the outside, and b) a peak on the inside.
• Thats a max pressure difference gt the eardrum
  is maximally stimulated.
• The cricket is happy!!

Trough
Time TP/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 gt MN synapses are temporarily depressed after
  the AN fires

Motor Neurons
MNR
MNL
ANR
ANL
Auditory Neurons
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

z1wi1
zi
Vi
z2wi2
Leak
z3wi3
EL
20
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
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 gt higher S

wij
S
t
zj
Vj
Vi
wij
zj
t
22
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
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.

• 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
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 ANLs signals are strong, MNL leaks
  too much and can never integrate enough charge to
  fire.

• This is cricket is very picky!