of 12
1
Supplementary Information for
2
Diversity enabled sweet spots in layered architectures and speed-accuracy trade-offs in
3
sensorimotor control
4
This PDF file includes:
5
Figs. S1 to S4
6
References for SI reference citations
7
1 of 12
In this supplementary document, we state the models and theories used to derive the results presented in the main text.
8
Robust control theory is used to characterize the performance of a feedback system using its input-output relation (
1
,
2
). The
9
fundamental limits are given in section 1.A for the basic model, and section 1.B for the layered model, followed by a survey of
10
relevant tools in the networked control theory in section 1.C. The neural signaling SAT is characterized by assuming a fixed
11
space and metabolic resources to build and maintain axons. The component SATs are derived for different encoding schemes:
12
spike-based in section 2.A and rate-based in section 2.B. Then, the system SATs are derived using the fundamental limits and
13
the component SATs for the basic model in section 3.A and for the layered model in section 3.B. Lastly, the experimental
14
detail and additional experimental results are presented in section 4.
15
Notations.
We use lower case letters to denote sequences,
i.e.
x
=
{
x
(0)
,x
(1)
,x
(2)
,...
}
, and
x
(
t
1
:
t
2
)
to denote a
truncated sequence of
x
from
t
1
to
t
2
,
i.e.
x
(
t
1
:
t
2
) =
{
x
(
t
1
)
,x
(
t
1
+ 1)
,
···
,x
(
t
2
)
}
. The
-norm of a sequence
x
is defined as
x
:=
sup
t
|
x
(
t
)
|
, and the
1
-norm of a sequence
x
is defined as
x
1
=
t
=0
|
x
(
t
)
|
. The mutual information between two
random variables
x,y
with the probability density function
P
is defined using
I
(
x
;
y
) =
E
[log(
P
(
x,y
)
/P
(
x
)
P
(
y
))]
. We use
Q
to denote a quantizer that approximates a continuous domain with a finite set of values. The quantizer is defined by
2
R
intervals partitioned by
{
p
[
`
]
}
and their respective representation points
{
c
[
`
]
}
such that
Q
(
x
) =
c
[1]
x
[
p
[0]
,p
[1])
c
[2]
x
[
p
[1]
,p
[2])
.
.
.
c
[2
R
]
x
[
p
[2
R
1]
,p
[2
R
]
)
,
[1]
where
R
is referred to as its data rate. We denote
̄
Q
R,
Ψ
to be a uniform quantizer with data rate
R
and domain
[
Ψ
,
Ψ]
, which
16
partitions its domain into
2
R
intervals with equal lengths and maps the input from each interval to the middle point of that
17
interval.
18
1. Fundamental limits in system performance
19
A. The basic model.
We consider the system with delayed and quantized control:
20
x
(
t
+ 1) =
x
(
t
) +
w
(
t
) +
u
(
t
)
[2]
21
where
x
(
t
)
R
is the system error,
u
(
t
)
R
is the control action, and
w
(
t
)
R
is the disturbance. We also assume zero
22
initial condition,
i.e.
x
(0) = 0
. Next, we describe the robust control problem and its solution in a deterministic setting and a
23
stochastic setting.
24
The feedback loop from sensor measurement
x
(
t
)
to control action
u
(
t
)
has a latency of
T
u
:=
T
s
+
T
i
with a signaling rate
R
. The delay
T
u
is composed of
T
s
, which models the nerve signaling delay, and
T
i
, which models other internal delays in
the feedback control loop (including both sensory and motor delays). The signaling rate
R
is defined to be the maximum
amount of information that can be transmitted by the control loop from sensors to actuators. The feedforward loop from
disturbance
w
(
t
)
to the control action
u
(
t
)
has an advanced warning of
T
a
, which allows the controller to get prepared for
future disturbance before it hits the system dynamics. Typically, the value of
T
a
depends on the speed of the rider and the
features on the trail. The total delay in control (
i.e.
delay from the moment the error dynamics are impacted by a disturbance
to the moment the control acts against the disturbance) is the latency minus warning:
T
:=
T
u
T
a
=
T
s
+
T
i
T
a
.
[3]
Deterministic worst-case.
In the worst-case setting, the disturbance
w
(
t
)
is assumed to be infinity-norm bounded. The
25
controller is characterized by the function
26
u
(
t
) =
K
(
x
(0 :
t
T
u
)
,w
(0 :
t
T
u
+
T
a
)
,u
(0 :
t
1))
.
[4]
27
We assume that the data rate
R
is minimum stabilizing,
i.e.
R >
0
(
3
). We consider minimizing the worst-case error normalized
28
by the size of the disturbance
29
inf
K
x
w
= inf
K
max
w
1
x
,
[5]
30
which is equivalent to the minimum values of
x
when the disturbance size is normalized to
w
= 1
. This problem admits
31
a simple and intuitive solution. In particular, the optimal cost is given by
32
max(0
,T
) +
(
2
R
1
)
1
,
[6]
33
where
T
from Eq. 3 is the total delay in control.
34
Remark 1
As we consider marginally stable systems, the system is minimum stabilizing when
R >
0
. However, in terms of
35
noise cancellation, the rate error and the total error go to
as
R
>
0
, so the actual system for sensorimotor control is
36
unlikely to be designed to be near the minimum stabilizing rate.
37
2 of 12
Remark 2
Eq. 6 states the size of the errors and informs how the performance degrades with delay. Moreover, Eq. 6 also
38
suggests the size of oscillations that can be induced by delays. In the experiment, if there is a large delay between the wheel and
39
the screen cursor location, and the bump’s disturbance has a higher frequency, we expect that the delay error will get amplified.
40
Oscillations in standing on one leg and in stick balancing can be easily tested by the readers.
41
Stochastic average-case.
In the stochastic setting, the signal
w
(
t
)
is independent and identically distributed Gaussian random
variables with zero mean and unit variance. The controller is characterized by the conditional probability density function
P
(
u
(
t
)
|
x
(0 :
t
T
u
)
,w
(0 :
t
T
u
+
T
a
)
,u
(0 :
t
1))
.
[7]
The communication in the feedback loop is done through an arbitrary discrete-time channel that satisfies the following
constraint:
lim
n
→∞
1
n
I
(
{
x
(0 :
n
T
u
)
,w
(0 :
n
T
u
+
T
a
)
}
;
u
(0 :
n
))
R.
[8]
Here,
R
is also assumed to be minimum stabilizing,
i.e.
R >
0
. On special case of Eq. 7 and Eq. 8 is to have a quantizer of
rate
R
in the feedback loop. The sensorimotor control in risk-neutral setting motivates us to consider an average error, and
and as such, our goal is to minimize the steady-state mean squared error normalized by error variance,
i.e.
inf
P
:
Eq.
8
E
[
x
2
]
E
[
w
2
]
,
[9]
which is equivalent to the following robust control problem:
inf
P
:
Eq.
8
lim
T
→∞
E
[
1
T
T
t
=1
x
(
t
)
2
]
.
[10]
This problem also admits a closed-form expression similar to its deterministic counterpart:
max(0
,T
) +
(
2
2
R
1
)
1
,
[11]
where the equality is attained by an additive Gaussian channel with capacity
R
. As a quantizer is a special case of Eq. 7 and
42
Eq. 8, the mean squared error cannot be smaller than the left hand side of Eq. 11. Recall from the main text that the first
43
term
T
in Eq. 11 can be considered as the delay error, the second term
(2
2
R
1)
1
can be considered as the rate error. The
44
delay error, the rate error, and the total error subject to the component SATs Eq. 27 is given in Fig. S2A.
45
B. The layered model.
Next, we consider the layered system with two feedback loops
46
x
(
t
+ 1) =
x
(
t
) +
u
(
t
) +
r
(
t
) +
b
(
t
)
[12]
47
The disturbance is now composed of two terms: a component
r
(
t
)
that is observed with advance warning
T
a
0
and a
48
component
w
(
t
)
that can be observed only through its impact on system response. We assume that the two disturbances are
49
bounded by
50
r
1
,
w
.
[13]
51
Here,
 >
0
captures the relative size of
b
(
t
)
in comparison to the size of
r
(
t
)
.
52
The controller is characterized by
53
u
(
t
) =
u
L
(
t
) +
u
H
(
t
)
u
`
(
t
) =
L
(
x
(0 :
t
T
`
T
c
)
,w
(0 :
t
T
`
T
c
)
,u
`
(0 :
t
1))
u
h
(
t
) =
H
(
x
(0 :
t
T
h
)
,r
(0 :
t
T
h
+
T
a
)
,u
h
(0 :
t
1))
.
[14]
54
The control action is generated by two nominally independent feedback loops, each having their own sensing, computation, and
55
communication components. Both feedback loops,
L
,
H
act through a motor nerve pathway with data rates
R
L
,R
H
and delays
56
T
L
,T
H
, respectively.
57
We consider minimizing the worst case inifnity norm of
x
(
t
)
,
i.e.
inf
H
,
L
sup
w
,
r
1
x
.
[15]
The optimal cost is given by
{
T
`
+
T
i
+
1
2
R
`
1
}

+ max(0
,T
h
T
a
) +
1
2
R
h
1
.
[16]
When
T
a
> T
h
, the value of Eq. (16) reduces to
{
T
`
+
T
i
+
1
2
R
`
1
}

+
1
2
R
h
1
.
[17]
3 of 12
C. Relevant tools from networked control literature.
Control under communication constraints has been extensively studied.
58
The comprehensive surveys (
3
6
) cover important issues in the field of networked control. The necessary and sufficient data rate
59
through the feedback loop in order to achieve system stability in linear stochastic control is studied in (
7
9
). Early study on
60
the relation between information and estimation accuracy with causal information structures appeared in the work of (
10
,
11
).
61
In the context of control, the optimal controller structure, separation principles, performance bounds are studied in (
10
,
12
27
).
62
2. Component SATs
63
In this section, we characterize the SATs for neural signaling in spike-based encoding and rate-based encoding. In a spike-based
64
encoding scheme, information is encoded in the presence or absence of a spike in specific time intervals, analogous to digital
65
packet-switching networks (28, 29).
66
A. Spike-based encoding.
We model the complex size distribution of axon bundles of identical radius. We use
T
s
,C
s
to denote
the delay and data rate (
i.e.
the amount of information in bits that can be transmitted) that can be communicated by the
axon bundles, respectively. When the signaling is precise and noiseless, an axon with the achievable firing rate
φ
can transmit
φ
bits of information per unit time. For sufficiently large myelinated axons, we assume that the propagation speed
1
/T
s
is
proportional to the axon radius
ρ
(30),
i.e.
T
s
=
α/ρ
[18]
for some proportionality constant
α
. We also model the achievable firing rate
φ
to be proportional to the axon radius
ρ
,
i.e.
φ
=
βρ,
[19]
for some proportionality constant
β
. Moreover, the space and metabolic costs of a nerve are proportional to its volume (
30
),
and given a fixed nerve length, these costs are proportional to its total cross-sectional area
s
. When the signaling is precise and
noiseless, the amount of information per unit time (bits/sec) that an axon with achievable firing rate
φ
can transmit is simply:
C
s
=
φ.
[20]
Combining above, we have
C
s
=
λT
s
.
[21]
where
λ
=
sβ/πα
is proportional to the spatial and metabolic cost to build and maintain the nerves.
67
B. Rate-based encoding.
In a rate-based encoding scheme, information is encoded in the spike rate. We can think of the
rate-based encoding as a Poisson-type communication channel whose input is the spike rate
γ
(
t
)
and the output is the
spike timing
M
(
t
)
. We assume that the spike timing is a non-homogeneous Poisson point process with rate (intensity)
γ
=
{
γ
(
t
)
0 :
t
R
+
}
, denoted by
P
t
(
γ
)
. The communication channel is then given by
M
(
t
) =
P
t
(
γ
)
.
[22]
where the spike rate is bounded by
γ
(
t
)
φ
t
R
+
,
[23]
for some
φ >
0
. The capacity of communication channel Eq. 22 is defined to be
C
r
= sup lim
T
→∞
1
T
I
(
γ
T
;
M
T
)
,
[24]
where the supremum is taken over all distributions of the input process
P
γ
(
t
)
satisfying Eq. 23. Kabanov has shown in (
31
)
that
C
r
is upper-bounded by
C
r
=
(
φ
+ 1)
1+
φ
1
2
(
1 +
1
φ
)
log(
φ
+ 1)
.
[25]
So for sufficiently large
φ
, we have
C
r
φ/
2
as
φ
→∞
.
[26]
which can be approximated by
C
r
=
λ
2
T.
[27]
4 of 12
Fig. S1.
The signaling rate in spike-based coding vs. rate-based coding given a fixed resource to build and maintain nerves. The solid black line shows the achievable signaling
rate
C
s
from Eq. 20 in spike-based coding; the solid blue line shows the achievable signaling rate
C
r
from Eq. 25 in rate-based coding; and the dotted blue line shows the
approximation
C
s
/
2
given in Eq. 27. The rate of the rate-based encoding is less than half of that of spike-based encoding and approaches to half of the spike-based encoding
rate as the achievable firing rate increases (see Eq. 26). This boost in signaling rate due to spike-based encoding may be particularly beneficial in highly constrained settings
(
e.g.
when the achievable firing rate
φ
is low, or when the available resource
λ
are limited).
5 of 12
C. Comparison of different encoding schemes.
Interestingly, the SATs of spike-based encoding and rate-based encoding are
68
qualitatively similar: given a fixed resource (space and metabolic cost to build and maintain a never), the achievable data rate
69
is roughly proportional to delay. The amounts of information that can be transmitted in the two encoding schemes subject to a
70
fixed resource are different and compared in Figure S1. The spike-based encoding allows more information to be transmitted
71
than rate-based encoding and thereby more efficient. Spike based encoding is particularly beneficial in the regime of low spike
72
rate and limited resources.
73
Although rate-based encoding has been believed to be the most standard encoding schemes conventionally, there is growing
74
evidence that nerves are able to use spike-based encoding (
32
). An important assumption behind spike-based encoding is that
75
axons are able to generate spikes in high timing precision. In this respect, existing literature has found that nerves are capable
76
of spiking with highly precise timing (
33
35
). Moreover, a few experiments also observe that spike timing carries behaviorally
77
relevant information (29, 34).
78
3. Optimizing system performance
79
A. System SATs in the basic model.
We can combine the fundamental limits in system performance from Section 1 and the
80
component SATs from Section 2 to obtain the system SATs. We showed the delay errors and the rate errors in control for two
81
cases: Eq. 6 in deterministic worse-case, and Eq. 11 in stochastic average-case. We derived the component SATs in two cases:
82
Eq. 21 for spike-based encoding, and Eq. 27 for rate-based encoding. We can combine these results to study different scenarios
83
and their system SATs.
84
Here we study the errors in the deterministic worse-case when spike-based encoding are used. Combining Eq. 6 and Eq. 21,
the optimal cost is given by
max
w
1
x
=
1
λ
R
+
T
i
T
a
+ (2
R
1)
1
if
1
λ
R
+
T
i
T
a
0
(2
R
1)
1
otherwise
.
[28]
Next, we compute the signaling delay
T
s
and rate
R
that minimizes the worst case error in Eq. 28,
i.e.
R
= arg min
R>
0
max
(
0
,
1
λ
R
+
T
i
T
a
)
+ (2
R
1)
1
[29]
T
s
=
1
λ
R
.
[30]
Let us define
C
(
R
) :=
1
λ
R
+
T
i
T
a
+ (2
R
1)
1
.
[31]
It can be shown that
dC
(
R
)
dR
=
1
λ
2
R
log(2)
(2
R
1)
2
.
[32]
The derivative
dC
(
R
)
/dR
with domain
R >
0
satisfies the following three conditions.
85
The function
dC
(
R
)
/dR
converges to negative infinity as
R
0
+
86
The function
dC
(
R
)
/dR
is increasing in
R
(
>
0)
.
87
If
C
(
R
)
has critical points,
i.e.
dC
(
R
)
/dR
= 0
, then the critical points satisfies
R
= log(
L
(
λ
))
[33]
where
L
(
λ
)
is defined to be
L
(
λ
) :=
1
2
{
(2 +
λ
log 2) +
4
λ
log 2 + (
λ
log 2)
2
}
.
[34]
The value of
L
(
λ
)
is increasing in
λ >
0
and
lim
λ
0
+
L
(
λ
) = 1
.
[35]
So for any
λ >
0
, the critical points Eq. 33 take a positive value.
88
From the above conditions, we obtain that
C
(
R
)
has a unique minimumizer (
i.e.
optimal signaling rate
R
) for any
λ >
0
.
This optimal rate is given by
R
=
{
log(
L
(
λ
))
if
1
λ
log(
L
(
λ
)) +
T
i
T
a
0
λ
(
T
a
T
i
)
otherwise
.
[36]
6 of 12
Combining Eq. 36 with the component SAT Eq. 21 yields the optimal signaling delay
T
s
=
1
λ
R
.
[37]
The preceding analysis can also be applied to other scenarios when Eq. 21 is replaced by Eq. 27 and/or when Eq. 6 are replaced
89
by 11. In these scenarios, there also exist a unique delay and data rate that minimizes the performance limits.
90
Beyond the component SATs in Eq. 21 and Eq. 27, the existence of a unique optimum depends on the specific forms of the
91
component SATs. On the other hand, Diversity-Enabled Sweet Spots can exist even if the delays and rate rates that minimize
92
the fundamental performance limits are not unique. Layering and diversity give more flexibility to optimize the delays and
93
data rates of individual layers according to the subtasks performed at each layer. To see this, we can consider a hypothetical
94
case that the optimal delays and rates is given by an interval (but not a point as in Fig 6B of the main text, the interval can
95
be either connected or not connected). As long as the optimal intervals do not have an intersection for the two layers, the
96
performance can be improved by layering and diversity.
97
Although there can exist some corner cases when the optimal intervals of the two layers have non-empty intersections, both
98
the theoretical prediction and empirical observations suggest that layering and diversity help improve the system performance
99
in the case studies in this paper: oculomotor control, balancing, and lateral control in humans.
100
B. Optimal component delay and data rate in the layered model.
Next, we consider the optimal delay and data rate of the
101
two-layer model in two scenarios: diverse cases vs. uniform cases. In the diverse case, the signaling delays and rates between
102
layers can be heterogeneous. In the uniform case, the signaling delays and data rates are homogeneous. The two cases are
103
assumed to have identical resource constraints to build and maintain the axons in both layers, which are quantified by the total
104
cross-sectional area to build axons.
105
We use
(
T
`
,R
`
)
and
(
T
h
,R
h
)
to denote the pair of signaling delays and data rates for the lower layer and the higher layer,
respectively. Both layers satisfy the component SATs
R
`
=
λ
`
T
`
[38]
R
h
=
λ
h
T
h
.
[39]
Recall from Section 2 that
λ
`
h
is proportional to the resource used in the lower layer and the higher layer, respectively. For
106
a fair comparison of the diverse case and the uniform case, we assume that both cases are under the same resource limitations.
107
So
λ
`
are assumed to be identical for both cases, and so does
λ
h
.
108
The optimal cost is given by
sup
w
δ,
r
1
x
=
{
T
`
+
T
i
+
1
2
R
`
1
}

+ max(0
,T
h
T
a
) +
1
2
R
h
1
.
[40]
We assume that the advanced warning
T
a
is sufficiently large,
i.e.
T
a

1
λ
h
log(
L
(
λ
h
))
,
[41]
where
L
(
λ
)
is defined in Eq. 34.
109
In the diverse case, minimizing Eq. 40 over the pairs
(
T
`
,R
`
)
and
(
T
h
,R
h
)
subject to Eq. 38 and Eq. 39 yields
R
`
= log(
L
(
λ
`
))
[42]
T
`
=
1
λ
`
log(
L
(
λ
`
))
[43]
and
T
h
=
T
a
T
i
[44]
R
h
=
λ
h
(
T
a
T
i
)
.
[45]
Here, we applied the results from Section 3.A.
110
In the uniform case, we additionally restrict the signaling delay to be homogeneous
T
`
=
T
h
.
[46]
As the delay is proportional to the cross-sectional area of the axons, this condition can be interpreted as having axons of
111
uniform size. For example, in the context of the oculomotor control system, this can be interpreted as the hypothetical setting
112
where the optic nerves and vestibular nerves have identical size (
i.e.
They are plotted at the same location in Fig. 2), as
113
opposed to optic nerves being orders of smaller and numerous. Note that Eq. 46 does not imply that the total delays in the
114
control loops are identical in the two layers since the total delays are also influenced by advanced warning and internal delays.
115
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Although the amount of performance improvement from the uniform case to the diverse case depends on the modeling
parameters, the worst case error of the diverse case is no greater than the worst case error of the uniform case. This can be
seen from
inf
T
`
,R
`
,T
h
,R
h
:
Eq.38,39
C
(
T
`
,R
`
,T
h
,R
h
)
inf
T
`
,R
`
,T
h
,R
h
:
Eq.38,39,46
C
(
T
`
,R
`
,T
h
,R
h
)
[47]
where
C
(
T
`
,R
`
,T
h
,R
h
) :=
{
T
`
+
T
i
+
1
2
R
`
1
}

+ max(0
,T
h
T
a
) +
1
2
R
h
1
.
[48]
Here, the left-hand side of Eq. 47 is the worst-case error in the diverse case, while the right-hand side of Eq. 47 is the worst-case
116
error in the uniform case. The constraint set of the right-hand side is a subset of the constraint set of the left-hand side.
117
Extensions and diversity with a layer.
In this paper, we focus on the role of diversity between layers relevant to the observed
118
nerve size and number heterogeneity (Fig. 2). The analysis can be easily extended to the case when
λ
h
and
λ
`
are also
119
optimization variables, which accounts for the varying resource use due to diverse nerve lengths. And related tools can be
120
used to study diversity within layers. For the theoretical and experimental study of diversity within layers, we refer interested
121
readers to our companion paper (
36
). The analysis methods presented in this paper and our companion paper can be combined
122
to consider a hybrid of diversity between layers and within a layer.
123
4. Experiments
124
A. Additional results for the multiplex experiments.
Recall from the Material and Methods in the main text that we tested
125
driving behaviors when there are curvatures in the trail, bumps in the road, and both. The first task mainly uses the reflex
126
layer, the second task the planning layer, and the third task both layers. The individual and combined errors are shown in Fig
127
4.
128
In addition, we measured the latencies of the two layers in these tasks as follows. We extracted the signals from
900
ms in
129
advance of the trail change or a bump to
1800
ms afterward. We measured the delay in each control task from the peak in the
130
cross-correlation between the time sequence of the disturbance (bump or trail change) and the time sequence of the control
131
input. To be able to compare with the trail effects, we flipped the sign of bump to make the cross-correlation positive. The
132
results for the first task, second task, and third task are shown in Fig. S4A, Fig. S4B, and Fig. S4C, respectively. The latencies
133
in the first and second tasks are estimated to be
1000
.
20
ms from Fig. S4A and
366
.
74
ms
from Fig. S4B, respectively.
134
Next, we further decomposed the errors in the third task into the error caused by the bump and the error caused by trail
135
changes. The decomposed signal for the trail effects and bump effects are shown in Fig. S4D and Fig. S4E, respectively. The
136
latencies in the estimated trail effects and bump effects in the third task are estimated to be approximately the same as the
137
first task (trail only) and the second task (bump only).
138
B. Test the impact of component SATs in control performance.
Recall from the Material and Methods in the main text that
139
we compared the driving behaviors when there are additional delay, quantization, and both from the steering wheel to the
140
trajectory. Other than the worst case framework described in the main text, we also tested the average errors in an average-case
141
framework, illustrated below. In the average case, we generated the turning angle (alternating to left or right) of the trail from
142
the uniform distribution with domain
[10
,
45]
. We quantified the system performance using the mean squared error between
143
the actual trajectory and the desired one. The measured errors for the three tasks are shown in Fig. S2B, which are consistent
144
with the theoretical prediction in Fig. S2A.
145
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146
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147
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