Hierarchical Network Model Excitatory-Inhibitory Tone Shapes Alternative Strategies for Different Degrees of Uncertainty in Multi-Attribute Decisions

Abstract

We investigated two-attribute, two-alternative decision-making in a hierarchical neural network with three layers: an input layer encoding choice alternative attribute values; an intermediate layer of modules processing separate attributes; and a choice layer producing the decision. Depending on intermediate layer excitatory-inhibitory (E/I) tone, the network displays three distinct regimes characterized by linear (I), convex (II) or concave (III) choice indifference curves. In regimes I and II, each option's attribute information is additively integrated. To maximize reward at low environmental uncertainty, the system should operate in regime I. At high environmental uncertainty, reward maximization is achieved in regime III, with each attribute module selecting a favored alternative, and the ultimate decision based upon comparison between outputs of attribute processing modules. We then use these principles to examine multi-attribute decisions with autism-related deficits in E/I balance, leading to predictions of different choice patterns and overall performance between autism and neurotypicals.

Full text

Hierarchical Network Model Excitatory-Inhibitory Tone Shapes
Alternative Strategies for Different Degrees of Uncertainty in
Multi-Attribute Decisions
Warren Woodrich Pettine1, Kenway Louie1, John D Murray2and Xiao Jing Wang1,*
1Center for Neural Science, New York University, New York, NY 10003, USA
2Department of Psychiatry, Yale University School of Medicine, New Haven, Connecticut
06510
*Corresponding Author. E-mail: xjwang@nyu.edu
January 28, 2020
Abstract
We investigated two-attribute, two-alternative decision-making in a hierarchical neural network with
three layers: an input layer encoding choice alternative attribute values; an intermediate layer of modules
processing separate attributes; and a choice layer producing the decision. Depending on intermediate
layer excitatory-inhibitory (E/I) tone, the network displays three distinct regimes characterized by linear
(I), convex (II) or concave (III) choice indifference curves. In regimes I and II, each option’s attribute
information is additively integrated. To maximize reward at low environmental uncertainty, the system
should operate in regime I. At high environmental uncertainty, reward maximization is achieved in
regime III, with each attribute module selecting a favored alternative, and the ultimate decision based
upon comparison between outputs of attribute processing modules. We then use these principles to
examine multi-attribute decisions with autism-related deficits in E/I balance, leading to predictions of
different choice patterns and overall performance between autism and neurotypicals.
1
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1 Introduction
We are constantly faced with decisions between alternatives defined by multiple attributes. The true value
of each attribute is at times clear, and other times uncertain. For example, on Friday one might choose
between main courses at a restaurant where the flavor or healthiness attributes of all the dishes are familiar.
The following Wednesday might be at a restaurant with an unknown cuisine, where one is highly uncertain
as to different items’ flavor or healthiness. To ensure the best meal, the brain must be able to optimize
choice in both environments.
Systems neuroscientists have, for many years, been studying the specific circuits engaged in this kind
of multi-attribute decision-making. Based on a robust set of electrophysiology and imaging findings (Xie
and Padoa-Schioppa 2016; Raghuraman and Padoa-Schioppa 2014; Padoa-Schioppa and Assad 2006; O’Neill
and Schultz 2018; Morrison and Salzman 2009; Conen and Padoa-Schioppa 2015; Chib et al. 2009; Pastor-
Bernier, Stasiak, and Schultz 2019), many hold that all attribute signals are available in brain areas proximal
to the final decision (Levy and Glimcher 2012; Padoa-Schioppa and Conen 2017). Indeed, when attribute
values are clear, multi-attribute choice theoretically is simple: linearly weight and combine all attributes
associated with a choice alternative, then select the one with the larger value. Though the subjective value
of an attribute might be non-linearly related to the quantity offered, when the final choice is made in an
environment without uncertainty, a weighted linear combination of attributes optimizes the choice between
options (Nicholson and Snyder 2007).
However, an alternative perspective has recently arisen that takes into account the multiple brain areas
implicated in decisions (Wunderlich, Rangel, and O’Doherty 2009; Peck, Lau, and Salzman 2013; Paton
et al. 2006; Munuera, Rigotti, and Salzman 2018; Louie and Glimcher 2010; Chen and Stuphorn 2015;
Steinmetz et al. 2019). Under this alternate view, a signal is transformed as it moves from hierarchically
lower areas to those that are proximal to the final decision (Cisek and Kalaska 2010; Cisek 2012; Yoo and
Hayden 2018). Such a process can allow for parallel computations, and produce clearer separation between
the representation of choice alternatives where the decision is reached. Furthermore, these transformations
can be highly non-linear, providing an additional layer of flexibility in the decision process.
When investigating the non-linear transformations that can occur in populations of neurons, it is imper-
ative to consider excitatory and inhibitory (E/I) tone. In sensory areas, E/I tone can shape the stimulus
tuning curve and responses timing (Mari˜no et al. 2005; Wehr and Zador 2003). When sensory information
is used in perceptual decision-making, E/I tone can dictate the trade-off between speed and accuracy (Wong
and Wang, 2006). In working memory tasks, E/I tone defines a network’s ability to maintain a memory, and
that memory’s susceptibility to a distractor (Brunel and Wang 2001; Compte et al. 2000). When multiple
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areas interact, the E/I tone can define their function in either memory maintenance or decision-making
(Murray, Jaramillo, and Wang 2017). On a more theoretical level, E/I tone can shape a system’s response
as an input signal increases (Ahmadian and Miller 2019) and its transition to chaotic dynamics (Vreeswijk
and Sompolinsky 1998). Yet, it remains to be described how E/I tone defines the functional interaction and
transformation of multiple signals passing through a brain area, as in a hierarchical system.
Though a prior biophysical model of hierarchical neural networks engaged in multi-attribute decision-
making captured key features of human reaction times and functional imaging (Hunt, Dolan, and Behrens
2014), how a hierarchical system shapes multi-attribute decisions is largely a matter of conjecture (Hunt and
Hayden 2017). By their very nature, these transformations distort the signals available for the final decision.
Distributed models have been shown to benefit motor control (Christopoulos, Bonaiuto, and Andersen 2015;
Cisek 2006), but in what situations might such transformations improve a decision process?
To approach this question, we examine how E/I tone in hierarchical networks governs multi-attribute
decision-making under varying degrees of environmental uncertainty. It is often the case that noise in
the environment creates uncertainty as to the true value of attributes (Bach and Dolan 2012). Additionally,
environmental uncertainty offers an entry-point to understanding neuropsychiatric conditions, such as autism
spectrum disorder (ASD). A myriad of findings suggest that the diagnostic signs of behavioral rigidity
and sensory abnormalities (American Psychiatric Association 2013) may be related to an intolerance of
environmental uncertainty (Foss-Feig et al. 2017; Fujino et al. 2017; Boulter et al. 2014; Van de Cruys
et al. 2014; Vasa et al. 2018). A mechanistic understanding of how hierarchical neural systems handle
environmental uncertainty when making decisions can not only provide knowledge as to how the brain
functions, but also may produce hypotheses as to the etiology of ASD.
We find that the E/I tone of the network creates distinct regimes defined by their choice indifference curve:
a linear weighting of attribute values (regime I); a convex preference for balanced attributes (regime II); or
a concave increased weighting of the larger attribute value (regime III). We then show that the degree of
reward maximization by these regimes depends on the level of environmental uncertainty. When uncertainty
is low, regimes I or II - where the choice area has access to all signal information - achieve the greatest
reward. However, when uncertainty is high, regime III - with strong transforms along the hierarchy favoring
the larger attribute value - on average, maximizes reward. After a detailed investigation of how these regimes
arise and why they are optimal, we hypothesize a framework where the E/I tone can be modified according
to the level of uncertainty. The framework is then used to create a model of neurotypicals with the full range
of E/I tone, and a model of ASD populations defined by limited inhibitory tone. Based on the models, we
predict that neurotypical and ASD subjects will adopt a linear weighting of attributes in environments with
low uncertainty (regimes I/II). As environmental uncertainty increases, neurotypical subjects will fully move
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into regime III, while ASD individuals will not. This will result in larger performances differences at high
environmental uncertainty levels.
Using a hierarchical network parameterized by E/I tone, we are able to show that the the network can
adopt multiple regimes (including linear) to maximize reward in various environments, and then we use these
findings to provide an etiological hypothesis for behavior observed in ASD.
2 Results
2.1 Networks Perform Multi-Attribute Decision Task Under Uncertainty
To investigate the behavior of neural networks in environments with varying degrees of uncertainty, we
used a multi-attribute decision task with varying degrees of environmental noise. On any given trial, the
neural networks were presented with two choice alternatives (Aor B). These choice alternatives were each
composed of two attributes (1 or 2). For simplicity, attributes were assumed to have equal contributions to
subjective value and actions were assumed to be symmetric. When indexing, the first value refers to the
choice alternative, and the second to the attribute. Thus, choice alternative Awas presented to the network
using input IA,1and input IA,2, while choice alternative Bwas presented to the network using input IB,1
and input IB,2. Inputs were translated from offer values to firing rates. As the final choice area competition
was between alternatives, the signal at that stage is denoted by CAor CB, where the subscript indicates the
choice alternative. On a given trial, the choice between Aor Bwas determined by the first population in
the choice area (CAor CB) to cross a firing rate threshold (35 Hz). To control the level of uncertainty in
the environment, we added a noise term ηIto each input at the beginning of the trial, independently drawn
from a normal distribution centered on 0, with a standard deviation of σηI.
We created two basic network frameworks: the Linear Network and the Hierarchical Network. Areas in
these networks were composed of mean-field approximations of population firing rates (see Methods). As
the name suggests, the Linear Network computed an exact linear weighted sum of the attributes (weight =
0.5). Thus, with the Linear Network, the choice area received a signal that was linearly translated from the
presented attributes of the choice alternatives (Figure 1A). This network has a similar architecture to that
presented in (Rustichini and Padoa-Schioppa 2015), and represents the case where all attribute signals are
fully available to a choice area.
In the Hierarchical Network (Figure 1B), inputs were first transformed by intermediate, attribute-specific
areas. The intermediate layer transforming the input signal was a nonlinear dynamical system, composed
of attractor states associated with each choice alternative (see Methods). To denote the transformation
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performed by an intermediate layer area on the input signal, its output signal will be referred to as T,
such that IA,1is related to TA,1, etc. These continuous output signals were then fed into a choice area
that determined the decision on a given trial. A single trial of the Hierarchical Network is shown in Figure
1C. This framework, similar in structure to that presented in (Hunt, Dolan, and Behrens 2014), represents
the simplest form of a network with parallel processing streams, where transformations are performed on
attribute signals passing through specialized areas prior to the final decision.
To first verify that the networks were able to perform the task, we computed simple psychometric functions
of model choice behavior. This was done by varying the input IA,1, while keeping inputs IA,2,IB,1, and IB ,2
fixed. We then measured the proportion of trials that networks chose A, and fit a sigmoid function to the
resultant P(A) (see Methods). We did this for environments where there was either no uncertainty (σηI= 0
Hz), or a moderate amount of uncertainty (σηI= 0.75 Hz).
Figure 2A shows the psychometric functions for the Linear Network in a certain environment (magenta)
and an uncertain environment (green). The same is shown in Figure 2B for the Hierarchical Network
with intermediate layer recurrent excitation (J+) = 0.33 nA and cross inhibition (J−) = −0.03 nA. Both
networks generated expected psychometric functions, with larger input value differences resulting in bet-
ter performance. Furthermore, the slope of the functions was, as anticipated, determined by the level of
environmental uncertainty, such that the greater uncertainty produced a less precise choice performance.
2.2 The Intermediate Layer Weights of Hierarchical Networks Dictate Decision
Regimes
Having verified the networks were able to perform the task, we next examined how they utilize attribute
information. While the Linear Network performs a simple linear sum of the attributes, the Hierarchical
Network first passes the input signals through attribute-specific areas (Figure 1B and Figure 3A). This
transform of inputs is defined by the recurrent excitation and cross-inhibition of areas in the intermediate
layer.
To investigate how the transform of attribute inputs changed with weights, we used indifference curves.
When generating an indifference curve for each weight-configuration, we varied the composition and magni-
tude of attributes in a single choice alternative ( IA,1and IA,2), while holding the composition and magnitude
of the other choice alternative (IB,1and IB,2) constant as a reference. For some values of IA,1and IA,2, (for
example when both are much less than their counterparts) P(A) will be low. For other values of IA,1and
IA,2, (for example when both are much greater than their counterparts) P(A) will be high. However, there
will be a set of values of IA,1and IA,2where the subject is indifferent between the choices and P(A)≈0.5.
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AB
C
IA,1 IB,1
IA,2 IB,2
IB,2
TA,2 TB,2
TA,1
CACB
CACB
Input
Choice
Transform
Choice
Transform
Input Input
IA,2
IA,1 IB,1 IB,2
IA,2
Time (ms)
FR (Hz)
Time (ms)
FR (Hz)
Time (ms)
FR (Hz)
Time (ms)
FR (Hz)
Time (ms)
FR (Hz)
Time (ms)
FR (Hz)
Lorem ipsum
TB,1
IA,1 IB,1
Figure 1: Network Schematics and Sample Trial. I: input; Cchoice layer synaptic gating variable; T: intermediate
transform layer synaptic gating variable; red: choice A; blue: choice B; black: attribute 1; white: 2; arrows:
excitation; circles: inhibition. Both networks include an input layer and a final choice layer. (A) The Linear Network
consists of two layers, with attribute signals from the input layer directly transmitted to the final choice layer. (B)
The Hierarchical Network includes an additional intermediate layer that performs a functional transformation on
the attribute signals prior to their passing to the choice area. (C) A sample trial of the Hierarchical Network, with
intermediate layer weights J+= 0.34 nA and J−=−0.02. For each area, the X-axis indicates time, while the Y-axis
indicates the population firing rate. The first vertical line indicates the onset of the offer value signal, and the second
vertical line indicates termination of the offer value signal.
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A B
No Environmental Uncertainty
High Environment Uncertainty
P(A)
(ÎA,1 + ÎA,2) - (ÎB,1 + ÎB,2)
Linear Network Hierarchical Network
Figure 2: Choice Performance Under Varying Environmental Uncertainty. The proportion of trials where the
network chose option A,P(A), as a function of the difference in offer inputs is shown, along with a fit sigmoid. Input
IA,1was varied as all other inputs were held fixed. On the x-axis, inputs were normalized ( ˆ
I) by the maximum choice
alternative offer value’s linear sum. (A) Performance of the Linear Network when attribute values are certain, with
σηI= 0.0 Hz (magenta), and when attribute values are uncertain, with σηI= 0.75 Hz (green). (B) The same is given
for the Hierarchical Network, with a intermediate layer weights of J+= 0.33 nA and J−=−0.03 nA.
The values where P(A)≈0.5 were used to fit the indifference curve.
The Linear Network by definition computes a weighted sum of the attributes, so its indifference curve is
trivially linear, following the line where IA,1+IA,2=IB,1+IB,2. For the Hierarchical Network, however, the
indifference curve provides a tool for analyzing how the network weighs and combines attribute information
(Figure 3). A linear indifference curve indicates that the network is performing a linear combination (similar
to that of the Linear Network). A convex indifference curve indicates that the network places greater weight
on attribute values composing a choice alternative that is balanced between the attributes, and less weight
on attribute values composing an unbalanced choice alternative. A concave indifference curve indicates the
opposite; specifically the network under-weighs the smaller attribute value, and instead place preferentially
greater weight on the larger of the attributes.
We parametrically varied the levels of excitatory (0.30 nA to 0.40 nA, spaced 0.05 nA) and inhibitory
weights (0.0 nA to −0.10 nA, spaced 0.05 nA) within intermediate layer areas. The varied weights are colored
yellow in Figure 3A. For each weight-configuration, we held the inputs IB,1and IB,2constant at 20 Hz, while
independently varying IA,1and IA,2from 0 to 40 Hz. We did this for 1,000 trials at each combination and
from those calculated P(A).
We then fit a single-parameter exponential function to the points where P(A)≈0.5 (see Methods). That
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B
C
A
Regime I
(Linear) Regime III
(Concave)
Regime II
(Convex)
IA,x IB,x
TA,x TB,x
ÎA,1
ÎA,2
Regime II
Regime III
Regime I
Inhibitory Weight (nA)
Excitatory Weight (nA)
J+ = 0.33, J- = -0.01 J+ = 0.34, J- = 0.0 J+ = 0.32, J- = -0.06
P(A)
Input
Transform
Figure 3: Decision Regimes of Hierarchical Networks. I: input; ˆ
I: input normalized by the reference choice
alternative values’ linear sum; T: transform layer synaptic gating variable; P(A): proportion of 1,000 trials where
alternative A was chosen; J+: excitatory weight; J−: inhibitory weight; yellow: varied weights; forest green: regime
II; tan: regime I; sky blue: regime III. The first subscript indicates choice Aor B, and the second subscript xindicates
a generic attribute. Weights were systematically varied identically in all intermediate layer areas. (A) A diagram of
a generic intermediate layer area and its inputs. (B) Examples of regime I (linear), regime II (convex) and regime
III (concave) decision regimes. (C) The space of E/I weights is partitioned according to the decision regime of their
indifference curves.
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single parameter determined the curvature, such that a value of 1 indicates a linear indifference curve, a value
less than one indicates convex, and a value greater than 1 indicates concave. We found that at different levels
of excitation and inhibition, the network adopted distinct decision regimes, corresponding to linear (regime
I), convex (regime II) and concave (regime III) indifference curves (Figure 3B). The appearance of these
decision regimes formed distinct regions in the weight-space, such that we were able to use these regions
to partition the weight-space (Figure 3C). This revealed that by configuring the levels of excitation and
inhibition in the intermediate layer, the Hierarchical Network can adopt different decision-making regimes.
2.3 Optimal Intermediate Layer Weights Depend on Magnitude of Environ-
mental Uncertainty
In what settings might distinct decision regimes in a hierarchical neural system be advantageous? Indeed, if
the goal is to maximize reward in an environment without uncertainty, a linear combination of attribute values
achieves optimality (Nicholson and Snyder 2007). Yet, many real-world decisions are made in environments
where the true value of an attribute is uncertain. For example, when examining a menu to choose a meal at a
restaurant, there may be dishes containing ingredients with which one is only vaguely familiar. We therefore
investigated the performance of these networks in environments where varying degrees of environmental noise
created uncertainty as to the true attribute values.
To do this, we sequentially presented networks with an array of 630 choice alternative combinations, which
were translated to firing rates of 10 to 20 Hz, spaced by 1 Hz. The inputs were produced combinatorially,
such that on one trial the Aattributes might be, [11 Hz, 12 Hz] and B, [13 Hz, 18 Hz]. On the next trial,
Acould be composed of, [11 Hz, 13 Hz] and B, [12 Hz, 18 Hz], etc. These choices were presented in blocks,
where each block had a fixed σηIthat ranged from 0 to 2 Hz.
For the Linear Network and each weight-configuration of the Hierarchical Network, we calculated the
average amount of reward-per-trial. Heatmaps of the reward-per-trial for the Hierarchical Network are
shown in Figure 4A, with the performance of the Linear Network indicated by the black dot on the heatmap
colorbar. The lower-bound of the heatmap colorbar indicates the minimum possible reward-per-trial (set to
0), and the upper-bound indicates the maximum possible reward-per-trial (set to 1). Weight-configurations
with regime I indifference curves are outlined in white. The region to the upper left of the white outlines
corresponds to regime II, and to the right to regime III. On the left of Figure 4A is performance in a
certain environment, and on the right performance in an environment with a high amount of environmental
uncertainty. In both heatmaps, the optimally performing weight-configuration is indicated by a black “X.”
The shift in the location of optimal weight-configuration is shown with more detail in Figure 4B, which plots
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A
DE
BC
σηI
= 1.5 Hz
Inhibitory Weight (nA)
Excitatory Weight (nA)
Reward Per Trial
σηI
= 0.0 Hz
Indifference Curvature
σηI (Hz)
Regime III
Regime I
σηI (Hz)
Excitatory Weight (nA)
Inhibitory Weight (nA)
σηI (Hz)
Reward Per Trial
Inhibitory Weight (nA)
Excitatory Weight (nA)
Uncertainty Slope
0.0
1.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
Regime I
Regime II
Regime III
Figure 4: Hierarchical Network E/I Tone Governs Reward Performance Under Varying Uncertainty. (A) Heatmaps
of the average reward-per-trial for different configurations of intermediate layer E/I weights. The X-axis is the
intermediate layer inhibitory weight, and the Y-axis is the excitatory weight. On the heatmap colorbar, the 0 value
is the minimum possible reward-per-trial, while 1 is the maximum possible reward-per-trial. The dot on the colorbar
indicates the performance of the Linear Network. Weight-configurations in decision regime I are outlined in white on
the heatmap. Regime II weight-configurations are to the upper left of those areas, and regime III are to the right.
The weight-configuration achieving the most reward is indicated by the black “X.” The left heatmap of (A) shows
performance in the absence of environmental uncertainty (σηI= 0 Hz), while the right heatmap shows performance
in the presence of high uncertainty (σηI= 1.5 Hz). (B) The optimal E/I weights shift as uncertainty increases. The
X-axis is environmental uncertainty (σηI), the left Y-axis shows the optimal excitatory weight and the right Y-axis
shows the optimal inhibitory weight. (C) The shift from regime I (tan) to regime III (sky blue, colors as in Figure
3C). The level of uncertainty is on the X-axis and the curvature of the fit to the indifference curve is on the Y-axis.
(D) A measure of the change in reward-per-trial as a function of noise for a regime I (J+= 0.34, J−=−0.01), regime
II (J+= 0.36, J−= 0.0) and regime III (J+= 0.35, J−=−0.10) weight-configuration. The magnitude of the slope
indicates the susceptibility to uncertainty. (E) The slopes of lines fit in (D) for all weight configurations. The slope
for the Linear Network is again indicated by the black dot.
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the optimal excitatory weight (magenta) and inhibitory weight (turquoise) across different magnitudes of
environmental uncertainty. Note that the optimal excitatory weight fluctuates in a narrow range (0.32 to 0.34
nA), while the optimal inhibitory weight magnitude increases monotonically (−0.01 to −0.1 nA). The shape
of the indifference curve produced by these weights is shown in Figure 4C, where the decision regime starts in
regime I (linear), and then moves to regime III, becoming increasingly concave as the environment becomes
more uncertain. Thus, the Hierarchical Network displays distinct decision regimes that maximize reward
under a variety of environmental uncertainty levels. Furthermore, the shift in regimes with uncertainty is
primarily achieved through increasing inhibitory tone.
We then measured how robust different weight-configurations of the Hierarchical Network were to changes
in the level of uncertainty. For each weight-configuration, we fit a line to the proportion of maximum reward
as the uncertainty level increased, as shown in Figure 4D. The magnitude of the slope was taken as the
susceptibility, with a greater slope indicating that the weight-configuration was less robust to uncertainty
changes. These slopes are shown in a heatmap on Figure 4E, with the slope of the Linear Network indicated
again with a black dot on the colorbar. We found that the regime III areas, in addition to displaying greater
performance with high uncertainty, were also more robust to changes in uncertainty levels.
2.4 Regime II Results from Nonlinear Summation, While Regime III Results
from Max-like Operation
Having established that the configuration of E/I weights defines decision-making regimes, and that these
decision-making regimes are optimal depending on the environment, we next examined the functional trans-
formation taking place within an intermediate layer attribute-specific area, and we analytically investigated
the conditions for which different regimes are optimal.
The phase planes of dynamical systems are a highly useful tool for analyzing the functional properties
of decision networks (Wong and Wang 2006; Wong, Huk, et al. 2007). For an intermediate layer area, each
combination of weight-configurations and input levels produced a unique phase plane. We calculated phase
planes for all combinations of an intermediate layer area’s excitatory weights (0.30 nA to 0.40 nA, spaced
0.05 nA) and inhibitory weights (0.0 nA to −0.10 nA, spaced 0.05 nA), and all input combinations of 0 to
40 Hz, spaced 0.5 Hz. This produced 793,881 total phase planes. For each phase plane, we computed a
noiseless trajectory starting near the origin. The endpoint of that trajectory was taken as the output of the
specific combination of network structure and inputs. The transformation of an input to output for different
weight-configurations is shown in Figure 5A, with the input on the left and the outputs of three different
weight-configurations on the right. We used these fixed point trajectory endpoints to approximate TA,1,
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TA,2,TB,1, and TB,2. The decision was determined by linearly summing these Tvalues to obtain CAand
CB, then choosing the maximum of these two choice values.
Though the intermediate layer outputs to the choice area in the full dynamical model often did not
reach the fixed points (due to the decision threshold), the fixed points did a reasonable job approximating
behavior. To measure this, for each weight-configuration we fit a sigmoid to the choice behavior of the full
model as a function of the difference between choice alternative fixed points (see Methods). Supplementary
Figure S2 shows the distribution of R2values. As over 90% of weight-configurations had an R2>0.70, we
felt confident in use of fixed points as an approximation for analysis of intermediate layer function.
AB
TA,x
Input Layer Transform Layer
J+ = 0.31, J- = -0.0
J+ = 0.34, J- = -0.01
C
TB,x
IA,x
IB,x
Inhibitory Weight (nA)
Excitatory Weight (nA)
TB,x Nullcline
TA,x Nullcline
Trajectory
T
ÎA,x- ÎB,x
A,x
B,x
Regime III
Regime II
Regime I
J+ = 0.32, J- = -0.06
Figure 5: Functional Properties of Hierarchical Network Decision Regimes. I: input; ˆ
I: input normalized by the
maximum input value; T: the transformed output of an intermediate layer; J+: excitatory weight; J−: inhibitory
weight; red: choice alternative A; blue: choice alternative B; green: trajectory from a starting point of (0.06, 0.06);
solid circles: stable fixed points; large solid circle: tra jectory endpoint. (A) The input to an intermediate layer
area is transformed to a fixed point output. The input values of IA,x = 15 Hz and IB,x = 10 Hz are presented
graphically as coordinates on the left. These inputs were fed into intermediate layer areas (arrows) with unique
weight-configurations. The resultant phase portraits of the transform layer are shown on the right and below for three
weight-configurations. (B) The weight-space was partitioned according to the decision regimes arrived at through
fixed point approximations. Indifference curves were computed for each weight-configuration by transforming offer
input values using the endpoints of their phase-portrait trajectories, summing them to compute choice values CAand
CB, then taking the larger of those values. The color scheme is the same as Figure 3C. (C) The input-output of a
single attribute value as it is varied (IA,x ) while the other (IB,x ) is fixed for a regime II weight-configuration (top)
and a regime III weight-configuration (bottom). The difference between the normalized fixed (ˆ
IB,x ) and the varied
value (ˆ
IA,x) is on the X-axis. The Tvalue of the endpoints is on the Y-axis. Individual TA,x (red) and TB,x (blue)
values are plotted, along with lines joining them to aid visualization.
We then used fixed points to compute indifference curves, similar to those shown in Figure 3B. In Figure
5B, we show the decision-making regimes as computed from the intermediate layer fixed points (as in Figure
3C). Its qualitatively similar form to that of Figure 3C further supports that the fixed points provide a
reasonable approximation of the full network.
Confident in the representativeness of the intermediate layer fixed points, we used them to investigate
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the input-output relationship of various intermediate layer weight-configurations. To do this, we first looked
at how the output changes as one input is increased while all else is held constant. We then assessed the
how multiple attributes interact as they pass through the intermediate layer.
To investigate how output changes as only one input is varied, we kept one of the attribute inputs (IA,x)
to an intermediate layer area fixed, while increasing the level of the other attribute input (IB,x ). The
resultant TA,x and TB,x for a region II weight-configuration (J+= 0.34 nA, J−=−0.01 nA), and a region
III weight-configuration (J+= 0.32 nA, J−=−0.06) are shown in Figure 5C. Note that in regime II (Figure
5C, top), the constant-input does not change as the magnitude of the variable input increases. Furthermore,
the output level of the varied-attribute (outlined in gray) in regime II shows a decreasing rate of change as
its magnitude grows. The same plot for regime III is shown in Figure 5C, on the bottom. In regime III, when
the variable input is less than the fixed input, the output-value of the variable input is suppressed; however,
when the variable input is greater than the fixed input, the output-value of the constant input is suppressed,
while the output-value of the varied input grows approximately linearly. Thus, in region I and II, each input
value passes through the intermediate layer whose degree of linearity is defined by the weight-configuration
(Supplemental Figure S3A). In region III, however, a qualitatively different function is implemented such
that the smaller input value is suppressed by a max-like operation (Supplemental Figure S3B).
Having established that different decision-making regimes maximize reward at different uncertainty levels,
and then how the network produces the regimes, the natural question arises as to why these regimes are
optimal in different environments. To answer this question, we turned to a mathematical analysis of simplified
non-dynamical models. One model represented the extreme case of sequential max-operations (corresponding
with regime III) and the other model that of a linear sum (regime I). We found that when the two choice
alternatives are extremely dissimilar, sequential max operations can lead to sub-optimal choices. However,
when the choice alternatives are similar, the max operations allow a network to improve the signal relative
to noise (see Supplemental Information for proofs).
2.5 Choice Patterns of Those With Autism will Differ from Neurotypicals as
Environment Uncertainty is Varied
Having shown that the Hierarchical Network can adopt multiple decision-making regimes, that the regimes
are optimal in different environments, and how E/I tone defines regime-specific attribute transforms, we
next hypothesized a framework where E/I tone can be modulated according to the degree of environmental
uncertainty. We then applied this framework to investigate altered decision-making in ASD.
ASD is a developmental disorder with strong evidence for inhibitory dysfunction (Foss-Feig et al. 2017;
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Inhibitory Weight (nA)
Excitatory Weight (nA)
Neurotypical Model
ASD Model
CD
σηI (Hz)
Indifference Curvature P(Larger Chose)
No Uncertainty
Medium Uncertainty
High Uncertainty
Trial Number
E
Food Per Trial
Regime III
Regime I
ÎA,1
ÎA,2
P(A)
σηI (Hz)
Neurotypical Model ASD Model
0.0
1.0
0.2
0.4
0.6
0.8
AUncertainty
Modulation
B
Intermediate Layer
M&Ms
Jelly
Beans
Basket A Basket B Basket A Basket B
Trial 1 Trial 150
...
Figure 6: Task Simulation with Neurotypical and ASD Models. (A) The two-alternative, two-attribute task with
varying levels of uncertainty. Alternatives, consisting of the attributes M&Ms and Jelly Beans, are simultaneously
presented. The offer value of an attribute is indicated by the position of the horizontal white line on the vertical
rectangle such that the higher the line, the greater the quantity of the attribute. Environmental uncertainty is
controlled through a yellow bar obscuring the exact location of the white line (right trial). (B) A module was added
(forest green) that modulated the intermediate layer E/I tone according to the uncertainty level. For the ASD model,
inhibitory tone was restricted to a maximum magnitude of −0.01 nA, as is shown by the magenta dots in intermediate
layer diagram. (C) A single session of the simulation, where the trial number is given on the X-axis and the proportion
of trials where the larger option was chosen given on the Y-axis. The smoothed value for the neurotypical and ASD
performance is given in black and magenta respectively. Blocks with no environmental uncertainty are in dark gray,
medium uncertainty in light gray, and high uncertainty in white. (D) The reward-per-trial as a function of uncertainty
for both the neurotypical and ASD models, with error bars drawn from sets of sessions. (E) The optimal indifference
curvature as a function of uncertainty for the neurotypical and ASD models. To the right are example indifference
curves for neurotypical (left) and ASD (right) at their maximum level of concavity. Input values (ˆ
I) are normalized
by the reference choice alternative values’ linear sum.
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Lee, Lee, and Kim 2017; Zikopoulos and Barbas 2013). Although E/I alterations observed in animal models
of ASD are highly diverse, one of the more consistent findings is alterations involving parvalbumin positive
(PV) interneurons, which have a strong influence on the overall inhibitory tone of a circuit. Gogolla et al.
found that PV interneurons were reduced across several areas of cortex and the hippocampus in a diverse set
of ASD mouse models (Gogolla et al. 2009). This selective reduction in PV interneurons was similarly found
in the cortex of human postmortem tissue, where only the proportion of PV interneurons was decreased
relative to controls (Hashemi et al. 2017). Decreases in PV populations are behaviorally relevant for ASD.
For example, mice specifically engineered for PV depletion display phenotypes strongly liked with ASD, such
as abnormal reciprocal social interactions, impairments in communication, and repetitive and stereotyped
patterns of behavior (W¨ohr et al. 2015). These and other findings have provided strong support for the
“E/I imbalance,” theory of autism, where inhibitory tone is restricted relative to excitatory tone (Foss-Feig
et al. 2017). Thus, manipulations of E/I tone in the hierarchical model can provide an avenue for linking
the biology of ASD with behavior.
In addition to restricted inhibitory tone in ASD, there is evidence for a general intolerance of uncertainty
(Boulter et al. 2014; Fujino et al. 2017; Vasa et al. 2018), and altered performance with environmental
uncertainty (Milne et al. 2002; Spencer et al. 2000). Furthermore, differences have been found in multi-
attribute decision tasks (Foxe et al. 2015; Zaidel, Goin-Kochel, and Angelaki 2015). We therefore created a
multi-attribute decision task with varying uncertainty as to the true value of attributes. The task, designed
for pediatric subjects, involves choosing between baskets of food for “Harry the Hippo.” Harry’s favorite
foods are M&Ms and Jelly Beans, which he enjoys equally. On a given trial, the network was presented with
two baskets to feed Harry, each composed of M&Ms and Jelly Beans (Figure 6A). In the figure, quantities of
the attributes are indicated by a horizontal lines on a vertical bars associated with each attribute. The goal
over the course of the session is to feed Harry as much food as possible. Uncertainty is created by obscuring
the exact position of the line with a yellow bar (right trial in Figure 6A); the wider the yellow bar, the more
uncertain the attribute’s true value.
The E/I tone of a brain area is not necessarily fixed. Indeed, there are several mechanisms by which the
tone can be temporarily modulated (Zucker 1989; Semyanov et al. 2004; Avery and Krichmar 2017). The
cholinergic diffuse modulatory system, in particular, provides a plausible mechanism for the source of E/I
tone shift due to expected environmental uncertainty (Yu and Dayan 2005).
To allow the network to adapt to uncertainty levels, we added a module that used a look-up table to
identify the optimal intermediate layer weight-configuration for a given environmental uncertainty level, then
shifted weights in the attribute-specific intermediate layers to that configuration (green in Figure 6B). The
neurotypical model was able to explore the full range of possible weights. The ASD model was implemented
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by limiting the possible inhibitory tone in intermediate attribute-specific areas to a maximum magnitude of
−0.01 nA.
Sessions were composed of trial blocks with variable levels of uncertainty. A single session is presented in
Figure 6C, with the smoothed proportion of the larger value chosen across trials shown for the neurotypical
(black) and ASD (magenta) models. Note that during high-uncertainty blocks (white), one can already see
a separation of performance levels. Analyzing food-per-trial as a function of uncertainty, we found that
the ASD model did not differ from the neurotypical model when the uncertainty level was low, but began
to diverge as uncertainty increased (Figure 6D). We also examined the change in the indifference curve as
a function of uncertainty. The trend is shown in the left portion of Figure 6E, while specific instances of
neurotypical and ASD indifference curves at the highest level are shown to the right. We found that there
was a ceiling to the concavity of the ASD model’s decision behavior. Thus, the model makes two strong
experimental predictions: 1) performance will be identical at low uncertainty, while subjects with ASD will
show a greater fall-off in total reward as uncertainty increases; and 2) at high uncertainty, subjects with ASD
will continue to show more linear-weighting of attribute values than neurotypicals, who shift to attending to
the larger of the attribute values.
3 Discussion
We showed that a hierarchical network of areas defined by E/I tone is capable of performing a multi-attribute
decision task, and can adopt behavioral regimes that maximize reward in a variety of environments. In
environments where attribution values are certain, the network maximizes reward by adopting a regime that
sums a linear translation of offer attribute values (regime I). However, as the environment becomes more
uncertain, the network maximizes reward by implementing a max-like function to filter noise (regime III).
We then proposed a framework where E/I tone can be modulated according to uncertainty. We used this
framework to predict that, due to an imbalance of inhibition relative to excitation, subjects with ASD will
diverge in behavior from neurotypicals as environmental uncertainty increases.
There is support in the literature for both the hierarchical and parallel structure of our model. Several
prior computational models have proposed a similar parallel processing of attributes that are then joined in
areas more proximal to the final decision (Roe, Busemeyer, and Townsend 2001; Balasubramani, Moreno-
Bote, and Hayden 2018; Hunt, Dolan, and Behrens 2014; Cisek 2006; Christopoulos, Bonaiuto, and Andersen
2015). Furthermore, recent studies involving humans directly examined attribute-specific processing prior to
integration in downstream areas. (Berker et al. 2019) looked at the representation of quantity and quality
attributes in gift cards. They found that distinct areas were independently specific for each attribute, and
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that a third area showed activity correlated with the combined signals from the earlier areas. The importance
of later-areas for the integration of value signals was demonstrated by (Pelletier and Fellows 2019) in patients
with damage to the OFC and vmPFC. They found that these patients were able to make decisions based on
single attributes (the form of elements in an object versus the configuration of elements) no different from
controls. However, when patients had to make decisions based on integration of the attribute signals, they
showed deficits. These results are expected from a neural system with parallel structure that converges for
integration of attribute information.
A down-stream area providing a common representation of attribute value information is a key component
of our model. Several electrophysiology studies have specifically investigated the kind of two-alternative
multi-attribute choices that we have examined, while recording from down-stream areas relevant for decisions
such as the OFC or ACC. Reward magnitude, probability, information value, social hierarchy and juice type
have all been found to be represented in these areas, and in a manner where all attribute signals were available
for decision-making (Hunt, Malalasekera, et al. 2018; Pastor-Bernier, Plott, and Schultz 2017; Pastor-Bernier
and Cisek 2011; Blanchard, Hayden, and Bromberg-Martin 2015; Cai and Padoa-Schioppa 2014; Raghuraman
and Padoa-Schioppa 2014; Cai and Padoa-Schioppa 2012; Xie and Padoa-Schioppa 2016; Padoa-Schioppa
and Assad 2006; Padoa-Schioppa 2009; Munuera, Rigotti, and Salzman 2018; Pastor-Bernier, Stasiak, and
Schultz 2019). However, it is important to recognize that the stimuli used to signify attribute information
in these studies were unambiguous, and that NHP subjects were over-trained on the use of attributes to
optimize reward. Thus, these studies all take place in the “low uncertainty,” range of our results. A
quasi-linear attribute value transmission is therefore expected. Our results suggest that if environmental
uncertainty is introduced, one should observe an over-representation of the larger attribute value in areas
such as the OFC or ACC, and that the activity will correlate with animal behavior. This is a distinct
prediction, highly feasible for experimentation.
Though the biology of how E/I tone of specific areas can be tuned according to task demands remains
to be fully characterized by experiments, diffuse modulatory neurotransmitter systems provide the most
plausible mechanism. Dopamine, norepinephrine, serotonin, nitric oxide and acetylcholine (ACh) have all
been shown to influence the E/I tone of the cells that they target (Avery and Krichmar 2017). These
influences can be complex, with cell-type specific shifts in excitation or inhibition implemented via a variety
of mechanisms (metabotropic receptors, ion channels, etc.). The ACh system in particular displays the area-
specific targeting (Zaborszky et al. 2015), the modulation of inhibitory tone (Disney, Aoki, and Hawken 2012;
Picciotto, Higley, and Mineur 2012; Herrero et al. 2008; Kang, Hupp´e-Gourgues, and Vaucher 2014; Phillips
et al. 2000; Sarter, Lustig, Howe, et al. 2014; Sarter, Lustig, Berry, et al. 2016; Thomsen, Sørensen, and
Dencker 2018; Disney and Aoki 2008; Disney, Domakonda, and Aoki 2006; Disney, Alasady, and Reynolds
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2014; Disney and Reynolds 2014) and the activity associated with known uncertainty (Monosov, Leopold,
and Hikosaka 2015; Voytko et al. 1994; Gill, Sarter, and Givens 2000; Dalley et al. 2004; Marshall et al.
2016) that is required by our model. However, it remains to be investigated precisely how the brain can
identify an task-optimal E/I tone for a circuit and then shift the circuit to that value.
Our work makes additional strong predictions for decisions of ASD subjects performing a multi-attribute
decision task as uncertainty is varied. There are a number of existing experimental results in the field of
ASD research that suggest our predictions are reasonable and worth investigating. Several studies show that
when attribute information is clear, performance between ASD and neurotypical subjects is identical, while
it differs in the face of noise or uncertainty. Dot field motion tasks, in particular, have been used to study
these questions in subjects with ASD. The general finding has been that when coherence of the moving dot
field is high (uncertainty is low), subjects with ASD and neurotypicals show similar performance (Milne et al.
2002; Spencer et al. 2000; Zaidel, Goin-Kochel, and Angelaki 2015). Indeed, in detection of velocity, and
some coherence studies, subjects with ASD outperform neurotypicals (Chen, Norton, et al. 2012). However,
when coherence levels decrease (uncertainty increases), subjects with ASD consistently show significantly
greater decreases in performance. Similar effects of environmental uncertainty have been shown in ASD
multi-attribute studies (Zaidel, Goin-Kochel, and Angelaki 2015; Foxe et al. 2015). Our results suggest a
plausible mechanism for the effects observed in prior studies, while also proposing a specific experiment with
falsifiable findings.
In this paper, we used a hierarchical neural network to investigate how biophysical properties, such as par-
allel structure and E/I tone, shape multi-attribute, multi-alternative decisions. This investigation produced
strong predictions for electrophysiology experiments, as well as experiments involving human behavior. Our
work thus suggests a new avenue for research connecting neural circuits to multi-attribute economic choice
and to deficits associated with autism.
4 Acknowledgments
This research was partly supported by the National Institutes of Health (NIH) grant 062349, and the Simons
Collaboration on the Global Brain program grant 543057SPI To XJW.
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5 Methods
5.1 Model Architecture
5.1.1 Cortical Areas
Areas in the networks were composed of mean-field attractors that have been used to model working memory,
perceptual decision-making, as well as to model individual areas in multi-area models of cortex (Wong and
Wang 2006; Wong, Huk, et al. 2007; Murray, Jaramillo, and Wang 2017; Mejias et al. 2016). Each area
consisted of two populations, such that c=A, B. The dynamics of a given population is described by a
single synaptic gating variable representing the fraction of activated N-methyl-D-aspartate receptor (NMDA)
conductance and governed by the equation (Wong and Wang 2006),
dS
dt =−S
τNM DA
+γ(1 −S)r, (1)
where the NMDA time constant τNM DA = 60 ms, and the rate of saturation is controlled by γ= 0.641. The
firing rate rwas a function of the input current I, as defined by the curve relation first described in (Abbott
and Chance 2005):
r=F(I) = aI −b
1−exp[−d(aI −b)],(2)
where a= 270 Hz/nA, b= 108 Hz, and d= 0.154 seconds. The total synaptic current consisted of recurrent
(Irec), noisy (Inoise ), background (Io= 0.3297 nA) and external components such that,
I=Irec +Inoise +Io+gIinput ,(3)
where the coupling constant g= 0.0011 nA/Hz converts the firing rate Iinput to current. As described in
section 5.1.2, specific values of Iinput are notated as IA,1,IA,2,IB,1, or IB,2. For a given population cin area
nof the the network, the recurrent current was given by the equation,
In
rec,i =X
m,j
Sm
jJ(m→n)
ij ,(4)
where J(m→n)
ij is the strength of the connection from population jin area mto population iin area
n. The use of this equation in the Linear Network and Hierarchical Network is described in the following
sections.
The noise currents for each population were defined by an Ornstein-Uhlenbeck process:
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τAMP A
dInoise(t)
dt =−Inoise(t) + ηI(t)qτAM P Aσ2
noise,(5)
where the time constant τAMP A = 2 ms, and ηIis a Gaussian white noise term with mean zero, and variance
σ2
noise = 0.003 nA.
5.1.2 Inputs
External inputs were provided as firing rates, which could span the range of 0 to 40 Hz. Notationally, they
are given by Iinput, such that the input of attribute 1 for choice alternative Awas written IA,1, etc. An
unspecified choice alternative is indicated with the population subscript “c” and and unspecified attribute
by the subscript “x.”
When varying environmental uncertainty, a value ηIwas randomly drawn from N(0, σ2
η) independently
for each attribute on each trial, such that ˜
Ic,x =Ic,x +ηI. The standard deviation σηIwas used to control
the amount of environmental uncertainty in a given block of trials.
In several figures, we refer to a normalized value of the input, ˆ
Ic,x. In those cases, the quantity is specified
with respect to which the input is normalized.
5.1.3 Linear Network
The Linear Network consisted of two layers. The first layer provided inputs from the individual attributes,
in the form of firing rates Iinput. The attribute inputs to a given population were linearly summed, such
that
Ic=w
a
XIc,a,(6)
where w= 0.5. These values were then fed into the choice area described in section 5.1.1. As there was only
one area, equation 4 simplified such that,
Ii=X
j
SjJij ,(7)
where for i=j, the connection strength was 0.3725 nA, and where i6=jthe connection strength was
−0.1137 nA (Wong and Wang 2006). The decision was determined when one of the choice area populations
passed the firing rate threshold of 35 Hz.
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5.1.4 Hierarchical Networks
The input layer to the Hierarchical Networks was also in Hz, and specified by Ic,a. In the Hierarchical Net-
work, however, inputs were segregated by attribute and fed into an intermediate transform layer, consisting
of a two-population area specific for each attribute. These areas were structured along the lines described
in section 5.1.1.
In the intermediate layer areas of the Hierarchical Network, equation 4 was defined as follows. J+
controlled the strength of the excitatory connection of a population to itself (A→A, etc.), and J−, the
strength of the inhibitory connection from one population to another (A→B, etc.). The strength of J
between areas in the intermediate layer areas was set to 0. Connectivity from the intermediate layer (IL)
areas to the choice layer (CL) was restricted to excitation between populations with the same selectivity,
such that J(IL→CL)
AA = 0.25, J(IL→C L)
AB = 0.0, etc.
The excitatory and inhibitory weights in this intermediate layer (J+and J−) were parametrically varied to
define the specific Hierarchical Network weight-configurations described in the Results section. For simplicity,
weights in all areas in the processing layer were symmetrically varied, such that when the recurrent excitation
or cross-inhibition were changed, all areas in the processing layer assumed the new values. The outputs from
this processing layer were dynamically fed into a choice area of the same type as the choice area of the Linear
Network in section 5.1.1. To clarify notation, when an input ris passed through an intermediate transform
layer area, its associated transformed output is referred to as T. The value Tis that of the synaptic gating
variable from equation 1.
5.2 Model Behavior
5.2.1 Psychometric Curves
For the psychometric curves shown in Figure 2, a single choice alternative attribute was varied while all
others were held constant. 1,000 trials were run for the Linear Network, and for the Hierarchical Network.
We then calculated the percent of those trials where the networks chose the option associated with the varied
attribute. To these points, we fit a sigmoid of the form:
P(c) = 1
1 + exp−k(Ic,x −µ),(8)
where P(c) represents the proportion of the time the varied choice alternative was chosen, kdictates the
slope of the sigmoid, Ic,x indicates the value of the varied attribute, and µprovides the centering of the
sigmoid.
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5.2.2 Indifference Curves
The indifference curves of Figure 3C and 5B were calculated by holding the values for choice alternative A
(IA,1and IA,2) constant as a reference, while independently varying IB,1and IB,2. The reference attribute
inputs were both 20 Hz, and the varied attribute inputs ranged from 0 Hz to 40 Hz. All possible combinations
of the varied attributes were then simulate for 1,000 trials, where the networks chose between the varied
alternative and the reference. This was done for the Linear Network, as well as for several processing-
layer weight-configurations of the Hierarchical Network. The processing-layer weight-configurations were all
possible combinations of the recurrent excitatory weights 0.30 to 0.40 nA, spaced 0.01 nA with the cross-
inhibitory weights 0.0 to −0.1 nA, spaced 0.01 nA. Indifference values were then taken to be those where
the network chose evenly between the alternatives.
To fit an indifference curve, we first normalized the indifference values such that the minimum in either
direction was 0, and the maximum was 1. We then fit the equation,
IA,2= (1 −Ia
A,1)1/a,(9)
where awas used to define the shape of the indifference curve. If a < 1, the curve was classified as convex.
If a= 1, it was classified linear, and if a > 1 it was classified as concave. These classifications were then
used in Figures 3C and 5B to partition the weight space into areas that produced convex, linear or concave
indifference curves.
5.2.3 Reward-Per-Trial
To simulate a choice experiment, we computed the choice behavior of the network in response to Ic,x values
in all possible combinations of the inputs ranging from 10 to 20 Hz, with a step size of 1 Hz, producing 630
possible combinations. This was done for the Linear Network, as well as for the Hierarchical Network with
all combinations of the recurrent excitatory weights ranging from 0.30 nA to 0.36 nA, spaced 0.01 nA, with
the cross inhibitory weights ranging from 0.0 nA to −0.1 nA, spaced 0.01 nA. Each choice was run for 1,000
trials, from which the percentage chosen was calculated.
Reward was considered to be the total value of the attributes for the chosen alternative, regardless if it
was the larger of the two. Thus, on each trial there was a maximum reward (the larger choice alternative) and
a minimum reward (the smaller choice alternative). The reward-per-trial was computed by simply averaging
the reward received across all trials. We did the same for the minimum and maximum reward. We then set
0 to be the minimum reward-per-trial and 1 to be the maximum reward-per-trial. Those values were used
to bound the colorbar in Figure 4A.
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5.2.4 Uncertainty Robustness
To compute the robustness to uncertainty, as shown in Figure 4E, we systematically stepped the σηIas
described in section 5.1.2 through the values 0 to 0.5 Hz with a step size of 0.1, as well as 0.75 to 1.0 Hz
with a step size of 0.25, and 1.5 to 2 Hz with a step size of 0.5. For the Linear Network and each weight-
configuration, the the reward-per-trial was calculated at each of these uncertainty levels. A linear function
was then fit to the reward-per-trial as a function of uncertainty level of the form,
RP T =aσηI+b(10)
Where RP T is the reward-per-trial, ais the slope and bthe intercept. The robustness to uncertainty
was assessed using the magnitude of a, such that the greater the magnitude, the more susceptible.
5.3 Fixed Point Analysis
To compute the fixed points and phase portraits of a processing layer area, we used the python package
pydstool (Clewley 2012), parameterized by the equations described in 5.1.1, with the internal noise was set
to 0. We first defined an attribute input as a coordinate in euclidean space space whose positions are defined
by the level of each input (Figure 5A, left). We then defined a phase plane by applying these inputs to each
weight-configuration. A starting position in each phase plane was selected near the origin at (0.06, 0.06),
and a noiseless trajectory was computed. The end point of that trajectory was taken as the output of the
processing layer.
5.3.1 Calculation of Convexity
The trajectory endpoints were computed for all inputs and weight-configurations described in section 5.2.2.
These endpoints were treated as the values T. They were then linearly summed to determine the value of a
choice alternative, such that,
Cc=X
a
Tc,x.(11)
The indifference point was taken to be when the CA≈CB. Convexity was then computed and the weight
space was partitioned using the methods described in section 5.2.2.
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5.3.2 Calculation of Functional I/O: Linearity
To determine the functional transformation of inputs as they passed through the processing layer, for each
weight-configuration we kept IB,x constant at 20 Hz while increasing IA,x from 0 to 40 Hz. We then recorded
the resultant TA,x and TB,x as the output of the functional transformation. We identified when the transition
of TA,x stabilized and fit the function,
TA,x =αIA,xh+β, (12)
where hwas used to determine the linearity of the transformation. An h≈1 indicated that the network
produced an output that was a linear function of the input. If h < 1 however, the network produced a
sub-linear output. The value of hwas plotted in the heatmap of Figure S3A.
5.3.3 Calculation of Functional I/O: Interaction
To calculate the full interaction of values, we computed TA,x as we varied both IA,x and IB,x from 0 to 40
Hz, with a step size of 0.5 Hz. We then plotted the values of TA,x shown in the heatmap of Figure S3B.
5.4 Neurotypical and ASD Models
We introduced a module that used a look-up table to assess the level of uncertainty in the environment, and
then set the weight-configuration in the intermediate layer areas to a tone that optimized performance. The
neurotypical model was able to utilize the full range of excitation (0.30 nA to 0.40 nA) and inhibition (0.0
nA to −0.1 nA). The ASD model was able to utilize the full range of excitation, but the inhibitory tone
was limited to a maximum of −0.01 nA. We then presented the models with blocks of trials at different
uncertainty levels, and calculated the average reward-per-trial (section 5.2.3) during each block.
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Supplemental Information
1 Analysis of Linear Model and Sequential Max Model
The Hierarchical Network achieves maximum reward in an environment without uncertainty by adopting a
weight-configuration that performed linear integration of attribute values (similar to the Linear Network),
while in an environment with high uncertainty a max-like operation achieved maximum reward. In this
section, we use simplified versions of the model to analytically investigate these observed phenomenon.
1.1 The Linear Model And the Sequential Max Model
We will consider and compare two models of multi-attribute decisions. In the linear model (LM), attribute
input values (I) for each choice alternative were linearly combined to reach a choice value CAor CB, and
then the maximum of those values was taken as the choice (as in Figure S1A). The sequential max model
(SM) consisted of a series of max operations (as in Figure S1B-C). These two models are extreme cases of
decision regimes adopted by the Hierarchical Network, as described in the main body of the paper. Though
we use the same notation as the main body of the paper for clarity (I,C, etc.), the values in this section
are unitless.
IA,1 + IA,2 IB,1 + IB,2
CACB
maxchoice(CA,CB)
=
=
max1(IA,1,rB,1) max2(IA,2,IB,2)
IA,1 + IA,2 0 + 0
CACB
maxchoice(CA,0)
=
=
max1(IA,1,IB,1) max2(IA,2,IB,2)
0 + IB,2
IA,1 + 0
CACB
maxchoice(CA,CB)
=
=
A B
Figure S1: Static Decision Models. (A) The linear model, where attribute input values (I) are summed into a
choice value (CAor CB). Choice is then determined with a max operation. (B) The sequential max model with
a max operation at the attribute level and the choice level. The surviving values from the first max operation are
summed to calculate C. Two examples of the sequential max model are shown, where the larger attribute input
values are bolded. On the left, both attribute inputs associated with Aare larger, and so these are used to calculate
the CA. Since the Battribute inputs are smaller, they do not survive the first stage, and so CB= 0. On the right,
an attribute associated with each choice alternative survives the first stage, and so both Cvalues are nonzero.
Our analysis will focus on the most basic case, where there are two choices composed of two attributes.
For notation, we will index the first choice as Aand the second choice as B. Note that the number of choices
could be expanded arbitrarily. We index the two attributes associated with each choice as 1 and 2. When
the two attributes are interchangeable, it will be indicated with the subscript x. An attribute could be the
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color of a choice alternative, the monetary value, or (if it is edible), its flavor. As an example, the input to
the model of the monetary value (1) associated with choice alternative 1 (A) could be written as IA,1. The
flavor (2) associated with choice alternative 1 (A) could be written as IA,2. We can then write the initial
offers as [IA,1, IA,2] and [IB,1, IB,2].
1.1.1 The Linear Model: Equations
The LM was meant to capture the Linear Network, along with the case of the Hierarchical Network model
performing a linear addition of the attribute values. The operations of the LM are shown in Figure S1A. In
this model, the attributes associated with each choice alternative are first summed, such that,
CA=IA,1+IA,2,and CB=IB,1+IB,2.(13)
The second decision stage then takes the maximum of these choice values as the choice of the network,
consisting of the operation,
maxchoice(CA, CB).(14)
For the linear model, we will often thus simplify the decision stage by writing,
maxchoice(IA,1+IA,2, IB,1+IB,2).(15)
1.1.2 The Sequential Max Model: Equations
The SM is named for the series of max operations performed, first at the input level, and then with the final
decision. Those operations are shown in Figure S1B-C. This model produces the concave decision behavior
described in the main text. The first stage of SM implements operations on the attribute level,
max1(IA,1, IB,1),
max2(IA,2, IB,2).
(16)
The key feature is that only the maximum of that operation is passed onto the next level. To gain intuition,
we can walk through a couple of examples. First, consider the case where,
IA,1> IA,2and IA,1> IB,2.
The decision process is shown in on the left of Figure S1B. Since only the winners of the first max
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operation (bolded in Figure S1) are passed to the next stage, the choice values will be,
CA=IA,1+IA,2,and CB= 0.
We can take another example to further illustrate the point (Figure S1C right). In this case, we are going
to define the values such that,
IA,1> IA,2and IA,1< IB,2.
After these values pass through equation 16, the choice values are,
CA=IB,1,and CB=IB,2.
Because of this, the Cvalues and composition will be specified in each examined case.
1.1.3 When Offers are Dissimilar, Max Produces Irrational Decisions
Here we will show that, in an edge case, the LM will choose the larger of the option, while the SM paradox-
ically chooses the lower.
We will consider the edge case where the value of Ais less than the value of B,
IA,1+IA,2< IB,1+IB,2.
We will further constrain the values so that,
IB,x =IB,1=IB ,2, IA,1< IB,x,and IA,1> IB,x.
In the LM, by substituting the values into equation 15, we can specify the decision stage as the operation,
maxLM,choice(IA,1+IA,2,2IB,x).
Given the constraints, the LM will choose B, which is the larger of the two offers.
In the SM, by use of equation 16, the first stage will consist of,
max1(IA,1, IB,x),and max2(IA,2, IB,x)
.
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Again, given the constraints, after the first stage we will have,
CA=IA,2,and CB=IB,x.
By use of equation 14, the decision stage will be,
maxSM,choice (IB,2, IB,x)
.
As IA,2> IB,x, the SM will choose A, even though the total value of Bis greater.
Conclusion 1 When the offers are highly dissimilar, there will be cases where the SM will choose the offer
with the smaller total value.
1.1.4 Uncertainty in the Linear Model and Sequential Max Model
In a biological system, uncertainty from noise can arise externally via the stimulus or value representation,
or internally via transmission between areas, or internally during the decision-making operation. For com-
prehensiveness, this section will consider how, at all stages, noise effects the LM and SM frameworks. Here,
we are agnostic as to the source of the noise (stimulus, representational, transmission, background, etc.),
and simply assume a random process, such that the noise term ηis drawn from a uniform distribution with
the limits [−ηmax,ηmax ]. By using a uniform distribution, we can more easily delineate the extreme cases.
When there are two stages at which the noise term is added, we will simplify their expression notationally
by writing,
ˆη=η1+η2.
we will consider the two-attribute, two-alternative decision paradigm. To make the algebra clearer, we will
introduce new variables D,Eand scaling term γsuch that,
D=IA,1, E =IB,1, γ E =IA,2, γD =IB,2.
We will also stipulate that,
D > E.
By maintaining the symmetry of the attributes across choice alternatives, we can finely control the magnitude
differences, while keeping the choice alternatives similar to one another. This can be better understood by
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examining the scaling term γ, which tunes the magnitude of the difference between the choice alternatives.
For clarity, we will first write rewrite equation 14 using the values as,
maxLM,choiceD+γE, E +γD.
It is easy then to see that the larger Cwill be dictated by the value of γ. If γ > 1, then Bwill be larger,
if γ < 1, Awill be larger, and if γ= 1 the total value of each choice alternative will be equal. Adding a
combined noise term, we arrive at,
maxD+γE + ˆηA, E +γD + ˆηB.(17)
If we define γas such that,
γD > E,
D > γE,
we can follow the algorithm, and arrive at the SM framework’s final decision stage where,
maxSM,choice D+ ˆηA, γD + ˆηB.(18)
The challenge then is to understand the relationship between scaling term γ(which controls the absolute
difference between choice alternatives) and noise term η.
1.1.5 Signal to Noise in the Linear Model Scale with the Difference Between Attributes
We will show that the noise term will be irrelevant in decision making as long as,
γLM >1 + ˆηmax
|IA,x −IB,x|(19)
Rewriting equation 17 so that Ais the winner we get,
E+γLM D+ ˆηA> D +γLM E+ ˆηB.
Giving alternative Bthe maximum benefit of noise we have,
E+γLM D−ηmax > D +γLM E+ηmax.
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Solving for the γLM terms on one side of the equation we get,
γLM (D−E)>(D−E)+(ηmax +ηmax).
By defining
ˆηmax = 2ηmax,
and dividing through by (D−E), and then back substituting the Ivalues, we arrive at equation 19. It
is important to see here that the minimum value of γLM depends primarily on the term,
ˆηmax
|IA,x −IB,x|.
The smaller this term, the smaller γLM can be.
Conclusion 2 The minimum possible value of scaling term γLM depends on the difference between IA,x
and IB,x. Thus, the larger the difference between the attribute values, the smaller the difference need be
between the choice alternatives.
1.1.6 Signal to Noise in the Sequential Max Model Scales with Magnitude of the Larger
Attribute
It can be shown that the value of γSM needed to render the noise irrelevant is given by,
γSM >1 + αˆηmax
max(IA,x, IB,x)(20)
Now, it is reasonable to consider the possibility that in a SM, the additional operation creates an addi-
tional source of internal noise. We will specify its scale using a variable αwhere,
α > 1.
We will start by rewriting equation 18 so that Awill be chosen over B. This can be specified by,
γSM D−αηmax > D +αηmax .
Solving for the γSM terms the equation becomes,
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γSM D > D +α(ηmax +ηmax ).
By dividing through by D, making the ˆηmax replacement, and back-substituting the Ivalues, we arrive
at equation 20. The important feature to note is that the minimum value of scaling term γSM depends on,
αˆηmax
max(IA,x, IB,x).
Unlike the LM, the denominator has only the larger of the attribute values.
Conclusion 3 In the SM, the larger the largest attribute value relative to the noise, the less the magnitude
difference between choice alternatives need be.
1.1.7 When Offers are Similar and in the Presence of Noise, Sequential Max Improves Signal
We will show that,
γLM −γSM >ˆηmax (Ia,clarger (1 −α) + αIa,csmaller )
Ia,clarger |IA,x −IB,x |,(21)
where,
Ia,clarger =max(IA,x, IB,x),
Ia,csmaller =min(IA,x, IB,x),
Ia,clarger =max(IA,x, IB,x),
Ia,csmaller =min(IA,x, IB,x).
We start by subtracting equation 20 from equation 19.
γLM −γSM >1 + ˆηmax
D−E−1 + aˆηmax
D.
By expanding the right side of the inequality we get,
γLM −γSM >ˆηmax
D−E−αˆηmax
D.
Putting both terms on the right side over the same denominator produces equation 21. Because D > E
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by definition, the denominator on the right will always be positive. However, for the inequality to hold, we
need to consider what must be true of the values of D,E,a,γLM and γSM . By doing so, we will be able to
draw some interesting conclusions regarding the two frameworks. First, lets consider the case where,
aE > D(1 −a).
This means that the numerator on the right side of the inequality will be positive. Thus for equation 21
to be true,
γLM −γSM >0,
and therefore,
γLM > γSM .
Conclusion 4 When the two attributes are similar, it will require a greater difference between the choice
alternatives (larger scaling term γ) for the LM model to recognize the more valuable choice alternative. If
one compares equation 19 with equation 20, in equation 19, the noise is divided by |IA,x −IB,x|, while in
equation 20 it is divided by max(IA,x, IB,x ). In the LM, the closer the attribute values, the less the noise is
reduced. In the SM, the smaller attribute value is no longer present; thus the noise term is always reduced.
Furthermore, if we consider the total noise in both models to be equal by setting,
a= 1,
then the numerator simplifies to,
ˆηmaxImin(A,B ),x,
which is always positive. In such a scenario, conclusion 2 will always be true.
1.1.8 If Internal Noise For the Sequential Max Model is Large, The Linear Model is Superior
Now, let us consider the extreme border-case where,
E= 0,
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and thus,
0>ˆηmax(1 −a)
max(IA,x −IB,x),
By itself this would say nothing about the value of γLM −γSM since it is possible that,
0> γLM −γSM >ˆηmax (1 −a)
max(IA,x −IB,x),
or,
γLM −γSM >0>ˆηmax (1 −a)
max(IA,x −IB,x).
Yet, as goal is the find the minimum possible value of the γs such that the inequalities hold, we can look
back to equations 19 and 20 for insight. In the border-case, equation 19 becomes,
γLM >1 + ˆηmax
max(IA,x −IB,x)
If minimize the value of γLM that would keep the inequality true, and assume that ais sufficiently large,
we get the inequality,
γSM >1 + aˆηmax
max(IA,x −IB,x)> γLM >1 + ˆηmax
max(IA,x −IB,x).
And thus,
γSM > γLM .
Conclusion 5 Due to the additional noise caused by multiple SM neural operations, if the difference
between the largest attribute and smallest attribute in each alternative is sufficiently large, the LM framework
will require less of a difference between the alternatives in order to choose the greater choice alternative.
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2 Supplemental Figures
Fixed Point CA- CB
J+ = 0.3 nA, J- = -0.04 nA
Network P(A)
R2
# Weight Configurations
Inhibitory Weight (nA)
Excitatory Weight (nA)
R2
ABC
Figure S2: Intermediate Layer Area Fixed Points Fit to Full Network Behavior. Fixed point trajectory endpoints
were used to approximate CAand CB. A sigmoid was then fit to the P(A) of the network as a function of fixed
point CA−CBfor each weight-configuration. The fit of a single weight-configuration is shown in A). An R2value
was then computed for each fit. A histogram of those R2values is given in B), and in C) they are organized by
weight-configuration.
Exponent
B
A
Inhibitory Weight (nA)
Excitatory Weight (nA)
(c)
(a)
(b)
IA,x
IB,x
Ta,1
J+ = 0.31, J- = -0.0 J+ = 0.32, J- = -0.06
Figure S3: Functional Transformations by Hierarchical Network Intermediate Layer Areas. (A) Polynomial functions
of the form TA,1=αIA,1h+βwere fit to the constant-regions outlined in gray on Figure 5C. The value of the h
exponent associated with each weight-configuration is indicated by the intensity of the heatmap. (E) The resultant
TA,x as IA,x and IB,x are varied for a regime I (left) and a regime III (right) weight-configuration.
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