System physiology – on the design
Petr Marsalek
Class: Advances in biomedical engineering
Graduate course, biomedical engineering
1
Outline, part 1
What is systems physiology;
Description levels:
Mathematics level;
Physics level;
Biology level
Design of the model;
(Case study 1 - ODE solver in Matlab, block design);
? Problems of reverse engineering;
Engineering design inspired by biology;
(Biomimetic engineering;
Neuromimetic engineering;
Bionics;)
2
Outline, part 2
Case study 2:
Internet atlas of physiology and pathological
physiology, demo.
Outline, part 3
Case study 3: Model of the flight control in
Drospohila Melanogaster (fruit fly)
Introduction to flight circuit;
Known facts;
Power muscles and steering muscles;
Neural circuitry, schematics of reflex arcs;
Why is feedback needed, the aerodynamics engineer's standpoint;
Design of the model;
(Methods - ODE solver in Matlab, block design);
Model tuning – sensory neurons emit one spike per wing cycle;
Left and right wing, amplitude and phase differences;
Modeling the saccade;
Exploring parameter space of one "linear" equation;
Limited options for the feedback and its function;
Towards comparison of model output with real data;
Concluding remarks
4
Neural sensori-motor circuits
5
Known facts
Neural circuits consist of neurons talking to each other through
synapses. Thoracic ganglion is a part of fly brain. Sensory inputs are
visual, mechanical and others (like odors etc.). Motor outputs are realized
by muscles. Motoneurons are last neurons in the circuit.
Most of the reflexes are fast (< 5 ms). Some of the reflexes are
monosynaptic.
Halteres – are a pair of club-shaped organs in a dipteran insect that
are the modified second pair of wings and function as sensory flight
stabilizers. Drosophila is an example of dipteran insect with one pair
of wings and with halteres. Compare e.g. to dragonfly of odonata with
two pairs of wings.
Flies have two types of flight muscles:
(1) power muscles and (2) steering muscles.
Experiments: (1) limited kinematics experiments: tethered flight, single
wing preparation; (2) behavioral experiments: free flight
Description of reflex arcs is based on anatomy of neural circuits.
6
Neural sensori-motor circuits
wing
descending
visual input
MN
SN
wing muscle
flight
flight
forces
flight trajectory
haltere
MN
SN
haltere muscle
7
Neural sensori-motor circuits
Reflex arcs of halteres
wing
descending
visual input
MN
SN
wing muscle
flight
flight
forces
flight trajectory
haltere
MN
SN
haltere muscle
8
Neural sensori-motor circuits
Reflex arcs of wings
wing
descending
visual input
MN
SN
wing muscle
flight
flight
forces
flight trajectory
haltere
MN
SN
haltere muscle
9
Function of Sensory Input
in Flight Control (Circuit)
Delay
Synapse
Sensory
neuron
Motoneuron
Mechanoreceptor
transduction
Mechanical
coupling
Muscle
Wing
Other inputs,
visual, from
halteres, etc.
Master
pacemaker?
No
Mechanical
Resonance?
Yes
Nonlinear
oscillator?
Yes
10
Model: Introduction of Leaky Integrator
(= RC circuit with threshold)
dV
 
VV
L
dt
  RC
gL  R1
V
(
t
dt
)

V
V

V
L for
TH
dV
g


g
(
V

V
)

I
L
L
L
dt
-50
-60
-70
right voltage [mV] V R
-80 left voltage [mV] V
50
52
54
56
58 L 60
62
time [ms]
t
64
66
68
70
Model: Leaky Integrator and
Spring Equations

dV
g


g
(
V

V
)

g
h
(
V

V
)

Np
(
x
)(
V

V
)

I
L
L
L
A
K
0
Na
dt
1
h(
V
)
1

V
(
t
dt
)

V
V

V
L for
TH SS


V

V
h
,half


1

exp
V 
dh
h
,slope


h 
h

h
(
V
)
SS
2
dt
d
x dx
m2


k
x

F
MRC
1
dt dt
p
(
x
)

0



E

k
x2
MRC

F  f (L)
1

exp
 kT 

B


Model: Reordered Equations

dV
g


g
(
V

V
)

g
h
(
V

V
)

Np
(
x
)(
V

V
)

I
L
L
L
A
K
0
Na
dt
V
(
t
dt
)

V
V

V
L for
TH
1. Although leaky integrator and spring
dh
h 
h

h
(
V
)
SS
dt
dx
y
dt
equations are linear, threshold,
adaptation and mechanoreceptor
currents are nonlinear, making the
whole DE set nonlinear.
2. Spring equation is rewritten to its
dy
normal form to be fed into a custom
m


y

k
x

F
MRC
written fixed step Runge-Kutta
dt
numerical DE solver (in Matlab).
Wings Model
Left and right wing is coupled through
variable stiffness K(t) to an oscillator =
oscillating power muscle
[Vilfan and Duke]
2
2
HE
x
x

F
y

B
(
x

y
K
t)xH
HH
HH
H
H)x
H
H(
2
2
HF
y
x

E
y

B
(
x

y
HH
HH
H
H)y
H
LxHw
w
K
t)xL
___
L
L(
Lw
x
L
RxHw
w
K
t)xR
___
R
R(
Rw
x
R
Wing amplitudes
and traces of leaky integrator
power oscillator x H
right amplitude x R
0.5
left amplitude x L
0
-0.5
50
52
54
56
58
60
62
64
66
68
70
right voltage [mV] V R
-80 left voltage [mV] V
50
52
54
56
58 L 60
62
64
66
68
70
-50
-60
-70
time [ms]
t
Wing amplitudes and phases
power oscillator x H
right amplitude x R
left amplitude x L
0.5
0
-0.5
50
52
54
56
58
60
62
64
66
68
70
right phase  R
left phase  L
0.5
0
-0.5
50
52
54
56
58
time
60
62
t [ms]
64
66
68
70
Feedback formula
K
(
t
)

g
(

(
t
)

0
.
6
),
for
t

t

t
L
F
L
i
i
i

1
KL(t) is time varying stiffness, gF is gain of the feedback,
L is wing phase. This is the formula for the left wing (L)
and analogous formula is for the right (R). [Tu and Dickinson, 96]
left amplitude
right amplitude
phase difference
amplitude difference
1
0.5
0
-0.5
55
60
65
70
75
80
85
90
95
left stiffness,  = 0.5 ms
right stiffness,  = 5 ms
alpha function,  = 0.5 ms
alpha function,  = 5 ms
0.4
0.2
0
-0.2
-0.4
55
60
65
70
75
time [ms]
80
85
90
95
Variables of a saccade
left amplitude, x L
right amplitude, x R
0.5
0
power oscillator, x H
-0.5
50
60
70
80
90
100
left phase,  L
right phase,  R
0.5
0
-0.5
50
60
70
80
90
100
left stiffness, K L
right stiffness, K R
0.5
0
-0.5
50
60
70
80
90
100
-50
left voltage, V L
right voltage, V R
-60
-70
-80
50
60
70
80
90
100
20
left relative spike timing
right relative spike timing
10
0
50
60
70
80
90
100
Saccade
19
[Fry et al, 2003]
Investigating parameter values:
without and with feedback
Amplitude
1.2
1
0.8
0.6
0.4
0.2
(magenta) amplitude, with feedback, varying gain
(red) amplitude, no feedback, varying K
0
0
0.5
1
1.5
2
2.5
3
gain g F, stiffness K
3.5
4
4.5
5
Investigating parameter values:
wing mass M=1 and M=2
No feedback, amplitude in dependence on stiffness
1.2
1
0.8
0.6
0.4
0.2
(blue) amplitude, mass M=2
(green) amplitude, mass M=1
0
0
0.5
1
1.5
2
2.5
3
stiffness K
3.5
4
4.5
5
From model to data
22
Function of Sensory Input in
Flight Control (Wish list)
Things to do, hypotheses, …
Theory: perturbation of the neural circuit will alter flight maneuvers.
Theory: test some of the popular hypotheses (eg. delay line in wing input).
Theory: what entrains/ perturbs wing rhythm?, phase lock, contributions…
Theory: minimal alterations of circuit, not possible in experiments.
Theory…
(Theory: any new ideas mostly sought and welcome…)
Theory: in general should (ideally) suggest interesting experiments.
Experiments: should (ideally) suggest interesting theoretical questions.
Experiments: calcium levels recording in mechanoreceptors and neurons.
Experiments: electrophysiological recording in mech.receptors and neurons.
Experiments: flight recording in mutants, in other Drosophila species.
Function of Sensory Input in
Flight Control (Wish list)
Things to do, hypotheses, …
V Theory: perturbation of the neural circuit will alter flight maneuvers.
+- Theory: test some of the popular hypotheses (eg. delay line in wing input).
V Theory: what entrains/ perturbs wing rhythm?, phase lock, contributions…
+- Theory: minimal alterations of circuit, not possible in experiments.
Theory…
(Theory: any new ideas mostly sought and welcome…)
X Theory: in general should (ideally) suggest interesting experiments.
+- Experiments: should (ideally) suggest interesting theoretical questions.
X Experiments: calcium levels recording in mechanoreceptors and neurons.
X Experiments: electrophysiological recording in mech.receptors and neurons.
X Experiments: flight recording in mutants, in other Drosophila species.
+-
Conclusions
1 The aim of the project is to understand the function of sensory
input in Drosophila flight control.
2 Equilibrium reflexes are described in experiments. Their
underlying circuitry is mostly unknown.
3 Current model: coupling of mechanoreceptors to spiking of their
sensory neuron. Closing of feedback loop from motoneuron to
sensory neuron.
4 We described the parameter space and key variables involved in
feedback and saccades.
6 What remains to do: to describe effects of feedback and steering
in terms of flight aerodynamics, which is the experimental
description level.
7 We will analyze new experimental data in near future.
25