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Evolution and analysis of minimal neural circuits
for klinotaxis in Caenorhabditis elegans
Overview
Spatial orientation is one of the most fundamental
adaptive behaviors in motile organisms.
C. elegans performs a diverse set of orientation
behaviors, including chemotaxis to tastants, odors,
osmolarity, electrotaxis, thermotaxis, etc.
Nearly all work on C. elegans chemotaxis was based
on klinokinesis (biased random walk).
Recently, Iino and colleagues have described a
complementary strategy, called klinotaxis.
This work combines neural network modeling and
evolutionary algorithms to identify simple circuit
motifs for klinotaxis. It then uses dynamical systems
analysis to understand how they function.
C. elegans as a model organism
Compact nervous system, with 302
neurons.
Full anatomical connectivity known.
Amenable to genetic manipulations and
electrophysiological analysis.
Microfluidics and fluorescent proteins are
allowing freely-moving worm recordings.
Neurons do not fire, they display graded
potentials.
Body and locomotion are well
understood.
Large behavioral repertoire, including
several forms of learning.
Chemotaxis in C. elegans
C. elegans chemotaxis is based in part on a biased random walk.
Recent finding
The worm curves during forward locomotion in the direction of the peak.
Constraints
Muscular contraction initiated at the
neck travels posterially. Each body
segment follows the one anterior to it.
Pair of sensory cells detect
changes in attractant
concentration (ON, OFF cells).
Dorsal / ventral symmetry.
Theoretical
challenge
attractant peak
worm
State-dependence.
Computational neuroethology approach
Idealised nervous
system and body
Population
Behavioral assay
on idealised environment
Recombination
Mutation
Dynamical system analysis
of successfully evolved networks
Idealized nervous system
N (1s)
ON
M (2s)
Pair of sensory cells detect changes in
attractant concentration:
OFF
wON
Two neck motor neurons (RMD),
modeled as CTRNN nodes.
wOFF
wS
VMN
wNMJ
Ventral
neck
muscle
DMN
wOSC
Oscillation
Dorsal
neck
muscle
Oscillatory element driving sinusoidal
body waves (but no CPG assumed).
The network is bilaterally symmetric
across the dorsal/ventral dimension.
No interneurons.
Idealized body
A
Head
Neck
(x, y)
v
Turning is proportional to the difference
between the two neck motor neurons.
μ
DMN
VMN
B
y
x
μi-1
Constant speed.
μi
φi Turning
angle
The path must contain convex and
concave curvatures, otherwise speed
drops.
White noise on the turning.
Idealized environment
Conical chemical gradient, not
Gaussian.
Fixed initial distance away from peak.
Random steepness.
4.5cm
Random initial orientation.
[0.1, 1.0] mM
Neural activity initialized to random
values.
Worm allowed to crawl for 500 secs.
Orientation changed at random with
some probability.
Each individual tested 50 times.
Results
Starting point
Gradient peak
300 sec
0 sec
1cm
1mm
Klinotaxis
Turning bias
(degrees)
Model
Bearing (degrees)
Worm
Evolved network motifs
8D parameter space.
A
Motif 1
10
Motif 2
0 θ
!10
10
wOFF
B
!10
VMN
DMN
Motif 1
0
!5
wS
10
wON
OFF
wOFF
VMN
θ
DMN
Motif 2
- Biases have different sign.
- Weights from sensor to motor
neurons also have different sign.
Sensory neurons with opposing
sensitivities always had antagonistic
synaptic effects.
ON
wON
OFF
wOFF
θ
!10
ON
wON
wS
0
5
Two distinct clusters:
The magnitude of the OFF weight
was on average 3 times stronger.
The self-connection was always
less than 4, which meant motor
neurons were not bistable.
Sensory dynamics
Decay time (sec)
A
B
Head sweep duration
4
c
Motif 1
Motif 2
3
yON
Head sweep duration
2
C
1
0
c
0
1
2
3
Rise time (sec)
4
yON
Head sweep duration
Two parameters define the dynamics
of the sensors.
On average, reaction times were
around the duration of a head sweep.
No systematic relationship between
motifs and sensory dynamics.
Turning bias modulated within an
individual head sweep.
Dynamical systems analysis, step by step.
1
2
3
4
4.5cm
[0.1, 1.0] mM
1
0
!1
π/2
0
D
Ventral
π
3π/2
Locomotion phase
B1
1
I
A1
I
Dorsal
0
!1
2π
D
V
σ(yD+θ)
Motif 2
A3
π
3π/2
Locomotion phase
2π
V
B2
1.0
1.0
0.8
0.6
0.6
σ(yV+θ)
0.8
0.4
0.4
0.2
0.2
0.0
!2
0.0
!2
!1
OFF
0
I
ON
1
2
1.0
B3 1.0
0.8
0.8
0.6
0.6
σ(yV+θ)
σ(yD+θ)
Motif 1
A2
π/2
0
0.4
0.2
0.0
!2
0.0
!2
ON
0
I
OFF
1
2
0
OFF I
!1
0
ON
1
2
1
2
0.4
0.2
!1
!1
ON
I
OFF
Strategy
Given that turning angle is proportional to the difference in
motor neuron output, we arrive at an extremely simple
strategy:
ON cell activation reduces turning angle.
OFF cell activation increases turning angle.
Interestingly, the exact same simple strategy
applies to both Motif 1 and 2, despite the
anatomical differences.
4
How do those simple rules at the level of the internal
dynamics lead to klinotaxis when embodied and situated?
4.5cm
Brain-body-environment interaction
ua
A
Upstep (3.0mM)
D
DS
a
VS
DS
b
u
c
V
d
x
uc
uc
Downstep (-0.75mM)
D
V
DS
a
b
VS
DS
u
c
d
y
x
2. ON reduces turning; OFF increases it.
Orientation response to single stepwise
changes in concentration at key phases
of the locomotion cycle.
y
B
1. Sensory response lasts around an
individual head sweep.
ua
The instantaneous velocity vector at
the time of an upstep signals the
direction of the implied peak.
Points b and d exemplify how crucial
the duration of the sensory dynamic is.
Generalized strategy
Upstep
A
D
a
60
DS
b
VS
c
d
DS
1. Turning response is a sinusoidal
function of where in the locomotion
cycle the step occurs.
2. The amplitude of the orientation
response is proportional to the
amplitude of the concentration step.
Turning bias
(degrees)
40
20
0
0.5mM
1.0mM
1.5mM
!20
!40
V !60 0
π
3π/2
2π
Locomotion phase
Downstep
B
D 60 a
DS
b
VS
c
d
DS
-0.5mM
-1.0mM
-1.5mM
40
Turning bias
(degrees)
3. The worm’s response to a downstep
of a given size is greater than the
response to an upstep of the same size.
π/2
20
0
!20
!40
V !60
0
π/2
π
Locomotion phase
3π/2
2π
Revisiting the theoretical challenge
Identical chemosensory signals sent
simultaneously to dorsal / ventral motor
neurons.
Effects are asymmetrical, allowing dorsal or
ventral turning bias.
attractant peak
Three key principles:
1. Biased motor neurons.
2. Antagonistic synaptic connections from
sensory neurons to motor neurons.
3. Timing of sensory dynamics related to headsweep duration, such that it cancels out at the
appropriate time.
worm
Ablation study
Model
Reliability
A
Reliability
B
1.0
0.8
0.6
0.4
0.2
0.0
Worm
Normal weights
Control
ASEL
ablation
ASER
ablation
Joint
ablation
Equalized weights
1.0
0.8
0.6
0.4
0.2
0.0
Control
ASEL
ablation
ASER
ablation
Joint
ablation
The role of the ON and OFF cells
Reliability
A
Reliability
B
1.0
0.8
0.6
0.4
0.2
0.0
Normal weights
Control
ASEL
ablation
ASER
ablation
Joint
ablation
Equalized weights
1.0
0.8
The reason for the difference in the
behavioral effect is not due to the OFF
cell being more strongly connected to
the motor neurons than the ON cell.
0.6
0.4
0.2
0.0
Control
ASEL
ablation
ASER
ablation
Joint
ablation
The functional asymmetry cannot be
fully deduced from the individual
components of the network.
The role of the environment on the role of
the ON and OFF cells
A
OFF cell only (ASEL ablation)
300 sec
0 sec
B
Starting point
Gradient peak
ON cell only (ASER ablation)
A simple geometrical argument is
sufficient to account for the
difference in effectiveness of the ON
cell and OFF cell during klinotaxis.
C. elegans chemotaxis specific predictions
1. Neck motor neurons need not be bistable.
2. Interneurons could be acting as passive conduits of activity.
3. Model suggests an antagonistic pathway between sensory and neck
motor neurons.
4. ON/OFF cell activation during forward locomotion should reduce/
increase the worm’s curvature (respectively).
4a. Turning should be sinusoidal function of locomotion phase.
4b. Turning should be bigger for OFF than for ON cell activation.
5. In ablation studies,
5a. Worms without ON cells should reach the gradient peak
reliably, and do so in long, spiral shaped trajectories;
5b. Whereas worms without OFF cells should reach the peak
infrequently, and do so with short, relatively straight trajectories.
General insights
Proposed mechanism and analysis may be useful in different C.
elegans behaviors (e.g., thermotaxis, electrotaxis).
Model provides an example of the same mechanism from
different network parameters.
Model provides an illustration of different roles for neurons
depending on geometry of environment.
Example of how modeling can be used as a cheap and quick
way to test the usefulness of an experiment.