Articles in PresS. J Neurophysiol (June 10, 2015). doi:10.1152/jn.00184.2015
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A Neuro-Computational Model of Economic Decisions
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Aldo Rustichini1 and Camillo Padoa-Schioppa2
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Department of Economics, University of Minnesota, 1925 4th Street South 4-101, Minneapolis,
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MN 55455. 2 Departments of Anatomy and Neurobiology, Economics, and Biomedical
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Engineering, Washington University in St Louis, 660 South Euclid Avenue, St Louis, MO 63110
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Running title: Wang's model for economic decisions
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Manuscript information: 250 words in Abstract; 750 words in Introduction; 10 Figures; 1
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Table
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Correspondence to: A.R. (
[email protected]) or C.P.-S. (
[email protected])
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Acknowledgments: This research was supported by the National Institute on Drug Abuse (grant
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R01-DA032758 to C.P.-S.) and the National Science Foundation (grant SES-1357877 to A.R.).
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We thank Nicolas Brunel, Xinying Cai, David Levine, John Murray, Xiao-Jing Wang and
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KongFatt Wong-Lin for helpful discussions, and Alberto Bernacchia, John Murray and Xiao-
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Jing Wang for comments on the manuscript. We also thank Xiao-Jing Wang and KongFatt
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Wong-Lin for sharing their Matlab code. 1 Copyright © 2015 by the American Physiological Society.
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Abstract
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Neuronal recordings and lesion studies indicate that key aspects of economic decisions take
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place in the orbitofrontal cortex (OFC). Previous work identified in this area three groups of
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neurons encoding the offer value, the chosen value and the identity of the chosen good. An
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important and open question is whether and how decisions could emerge from a neural circuit
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formed by these three populations. Here we adapted a biophysically realistic neural network
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previously proposed for perceptual decisions (Wang 2002; Wong and Wang 2006). The domain
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of economic decisions is significantly broader than that for which the model was originally
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designed: yet the model performed remarkably well. The input and output nodes of the network
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were naturally mapped onto two groups of cells in OFC. Surprisingly, the activity of
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interneurons in the network closely resembled that of the third group of cells, namely chosen
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value cells. The model reproduced several phenomena related to the neuronal origins of choice
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variability. It also generated testable predictions on the excitatory/inhibitory nature of different
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neuronal populations and on their connectivity. Some aspects of the empirical data were not
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reproduced, but simple extensions of the model could overcome these limitations. These results
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render a biologically credible model for the neuronal mechanisms of economic decisions. They
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demonstrate that choices could emerge from the activity of cells in the OFC, suggesting that
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chosen value cells directly participate in the decision process. Importantly, Wang's model
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provides a platform to investigate the implications of neuroscience results for economic theory.
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Introduction
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Economic choices are thought to entail two mental stages: subjective values are first assigned to
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the available options, and decisions are made by comparing values. Evidence from lesion studies
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(Fellows 2011; Rudebeck and Murray 2014), functional imaging (Bartra et al. 2013; Clithero and
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Rangel 2013) and neurophysiology (Mainen and Kepecs 2009; Padoa-Schioppa 2011; Wallis
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2011) indicates that choices, in particular choices between goods, involve the OFC. Neuronal
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recordings in primates choosing between different juices identified three groups of neurons in
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this area: offer value cells encoding the value of individual juices, chosen juice cells encoding the
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binary decision outcome, and chosen value cells encoding the value of the chosen juice (Padoa-
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Schioppa 2013; Padoa-Schioppa and Assad 2006). Prima facie, these groups of neurons appear
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sufficient to characterize ‒ and possibly generate ‒ a decision. Indeed, offer value cells capture
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the input to the decision process, while chosen juice and chosen value cells capture the identity
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and value of the chosen good, and thus the decision outcome. A primary goal for decision
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neuroscience is to formalize this intuition by building a biologically realistic model in which the
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groups of cells found in OFC form a circuit that generates decisions. Ideally, such a model would
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encompass all that is known about these neurons and, concurrently, make new and testable
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predictions.
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Our thinking about the mechanisms of economic decisions is influenced by work on motion
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perception (perceptual decisions). In a simplified scheme, two brain regions play a primary role:
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the middle temporal (MT) area, where neurons encode the momentary evidence, and the lateral
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intraparietal (LIP) area, where cells represent the decision outcome in the form of a planned
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saccade (Gold and Shadlen 2007; Parker and Newsome 1998). Notably, there is a natural
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analogy between neurons in MT and offer value cells, and between neurons in LIP and chosen
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juice cells. In contrast, chosen value cells do not have a known correspondent in perceptual
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decisions.
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Several models have been proposed to describe the neuronal mechanisms of perceptual decisions
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(Bogacz et al. 2006; Drugowitsch and Pouget 2012; Gold and Shadlen 2007). At the biophysical
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level, a leading proposal is Wang's model, in which decisions emerge from a balance of recurrent
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excitation and pooled inhibition. Different variants of the model account for perceptual decisions
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(Wang 2002; Wong and Wang 2006), similarity judgments (Engel and Wang 2011), probabilistic
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inference (Soltani and Wang 2010), behavior in a competitive game (Soltani et al. 2006) and
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flexible sensori-motor mapping (Fusi et al. 2007). More recently, the model has been adapted to
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describe the activity in LIP during foraging tasks (Soltani and Wang 2006) and to fit aggregate
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neural activity during value-based decisions (Hunt et al. 2012; Jocham et al. 2012). However, a
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precise mapping between Wang's model and the activity of neurons in OFC (or any brain area)
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during economic decisions has not yet been attempted. In this study, we examined the extent to
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which Wang's model can reproduce neuronal activity in the OFC. From a modeling perspective,
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this is a challenging test because – as we argue in detail below – the domain of economic 3
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decisions is significantly broader than that for which the model was originally designed, and
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because neuronal activity in the OFC during economic decisions only partly resembles that in
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area LIP during perceptual decisions. Our deliberate goal was to test the model without changing
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its structure. Importantly, all the parameters in the model represent biophysical quantities such as
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synaptic efficacies and time constants and are derived, at least approximately, from empirical
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measures (Wang 2002). In our investigation, we used the same parameters previously used to
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model perceptual decisions (Wong and Wang 2006).
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In most respects, Wang's model reproduced the activity of different groups of cells in OFC
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remarkably well. The two layers corresponding to the input and to the categorical output of the
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network were identified with offer value cells and chosen juice cells, respectively. Most
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surprisingly, the activity of inhibitory interneurons in the network closely resembled that of the
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third group of neurons found in OFC, namely chosen value cells. The model also reproduced
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several phenomena related to the neuronal origins of choice variability, namely choice hysteresis,
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the "predictive activity" of chosen juice cells and the "activity overshooting" of chosen value
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cells (Padoa-Schioppa 2013). Two aspects of the empirical data were not reproduced. First, the
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model did not include neurons with negative encoding. Second, a significant baseline in the
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activity of offer value cells introduced distortions in the behavior of the network. However,
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simple extensions of the model could overcome these limitations. The results of this study render
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a biologically credible model for the neuronal mechanisms of economic decisions.
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Materials and Methods
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Structure of the model
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In its extended form (Brunel and Wang 2001; Wang 2002), the model is a recurrent network of
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2,000 spiking neurons, of which 80% (NE) are excitatory pyramidal cells and 20% (NI) are
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inhibitory interneurons. All neurons in the network are leaky integrate-and-fire cells endowed
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with biophysically realistic parameters. Two external stimuli provide the primary input, and each
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stimulus activates a fraction f = 0.15 of pyramidal cells. The remaining (1-2f) NE pyramidal cells
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are non-responsive (untuned). The synaptic input to each neuron is both excitatory and inhibitory. 4
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Excitatory inputs are through AMPA- and NMDA-mediated synapses, while inhibitory inputs
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are through GABAA-mediated synapses. For each neuron, the input has an external component
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and a recurrent component. For the two groups of selective pyramidal cells, the external
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component includes the external stimulus. In addition, each neuron in the network receives an
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external background noise distributed as a Poisson process. Stimulus current and external
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background are through AMPA-mediated synapses. The recurrent component is provided by
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other neurons in the network. In analogy to the Hebbian rule, synapses between neurons in a
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given group (which fire together) are potentiated by a factor w+>1, whereas synapses between
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neurons in different groups are depressed by a factor w–1) discussed above. In essence, we propose that the network responds to
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the changes in value ranges by altering the synaptic efficacies between OV cells and CJ cells
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(Eq.19). As previously shown, this synaptic plasticity can be achieved with a mechanism of
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Hebbian learning (Padoa-Schioppa and Rustichini 2014). Framing effects may be reproduced if
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this Hebbian learning lags range adaptation of OV cells.
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Stable points of the dynamical system
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One important question concerns the number steady states of the model. Wong and Wang (2006)
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provided a bifurcation diagram, in which the number of attractors and the working-memory
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regime were examined as a function of the input firing rate (μ0) and the parameter w+. Their
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analysis was done for the reduced version of the model (W2) and for coh = 0. Generating a
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bifurcation diagram for W11 is more difficult because the system is time dependent and high
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dimensional. As a first step, we conducted a series of simulations to examine the end state of the
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network for different sets of offer values (Fig.9). For these simulations we set ΔA = ΔB = [0 15]
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and δJstim = (1, 1), such that ρ = 1. For comparison, although w+ defined in W11 is not identical
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to w+ defined in W2, the value of w+ used here is comparable to values examined by Wong and
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Wang (2006). In contrast, stimulus currents are substantially higher in our case. Specifically, in
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our simulations we normally used ΔJ = JAMPA,input / JAMPA,ext,pyr = 30 (see Table 1). In such
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conditions and for the data point at the center of Fig.3 (right panel), the product JAMPA,input rOV is
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roughly equal to 100 JAMPA,ext,pyr (see Eq.19). In contrast, in the original W11, the equivalent
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product was set to 40 JAMPA,ext,pyr when coh = 0.
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Consider Fig.9, panel a2. In this scatter plot, x- and y-axes represent, respectively, the "final"
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activity of CJA and CJB cells as measured in the time window 400-600 ms after stimulus on.
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Each symbol + represents one trial and symbols are color coded according to the offer type. The
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color legend is indicated in the leftmost panel in the same row. Rows 2 and 3 of the figure
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illustrate the same data focusing on a subset of trials, with the color legend indicated in the
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leftmost column. Several points can be noted. First, the network clearly separated between the
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two end points when the value difference was sufficiently high (purple, blue and green symbols
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in panels b1 and b2). It did not clearly separate between the end points when offer values were
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close (panel b3). In separate simulations, we observed that extending the stimulus in time for 1 s
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or increasing the levels of noise in the system did not affect these results (not shown). The 22
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simulation in panels b1-b3 were based on our normal parameters w+ = 1.75 and ΔJ = 30. We then
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repeated the simulations using parameters w+ = 1.85 and ΔJ = 30 (panels c1-c3), and w+ = 1.75
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and ΔJ = 15 (panels d1-d3). The results obtained were qualitatively similar, although the
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separation for difficult decisions tended to increase when w+ was raised (panel c2) and to
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decrease when ΔJ was reduced (panel d2).
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The formulation of W2 for economic decisions and a more formal analysis of the steady states
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will be presented elsewhere.
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The baseline activity of offer value cells
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As illustrated in Fig.1a, offer value cells presented a baseline activity of ~6 sp/s, which is smaller
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but comparable to the value-related dynamic range (~10 sp/s). Experimental work showed that
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this baseline activity does not depend on the value range, on the juice preference or, for given
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juice pair, on the relative value of the juices (Padoa-Schioppa 2009). As we examine the effects
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of introducing a baseline in the activity of OV cells, two premises are in order. First, the original
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W11 set the baseline equal to zero – a reasonable approximation since the baseline activity of
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MT cells is indeed low. Second, the presence of a substantial baseline changes dramatically the
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input to the network.
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In a series of simulations, we tested W11 introducing a realistic baseline. If we consider the
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situation in which all synaptic efficacies are balanced, the network is robust to the introduction of
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a baseline. In this condition, even using the same parameters as in the initial simulations, we
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obtained for the indifference function a straight line through the origin (not shown). Small
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adjustments to the parameters provided realistic profiles for CJ cells and for CV cells (Fig.10a-f).
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However, problems arise when one considers the fact that the relative value ρ between two goods
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should be free to assume any value (see above section Imposing non-trivial relative values)
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and/or the fact that (only) the value-dependent component of offer value cells activity adapts to
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the range of offer values (see above section Range adaptation, context-dependent preferences
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and Hebbian learning). The approach used in the baseline-subtracted case, namely to introduce
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an imbalance in the synaptic efficacies no longer works. In other words, the indifference function
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becomes non-homogenous even if the synaptic imbalance is limited to the input (Fig.10g-l). This
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is because the synaptic efficacies δJstim multiply the entire activity of OV cells, including the
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baseline. As a consequence, when δJstim, 1>δJstim, 2, the network chooses zero quantities of juice A
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over small quantities of juice B. In conclusion, the baseline activity of offer value cells poses a
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challenge for W11. Possible ways to address this issue are discussed below.
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Discussion
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We examined a biophysically realistic model previously proposed to describe the activity of area
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LIP during perceptual decisions, namely W11, and we tested the extent to which it could
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reproduce the activity recorded in the OFC during economic decisions. Our analysis represented
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a challenging test for the model for at least four reasons. (1) Stimuli (i.e., offers) in economic
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decisions are intrinsically two-dimensional, whereas stimuli in perceptual decisions are one-
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dimensional. (2) The relative value between two given goods is arbitrary. Thus to accommodate
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subjective preferences, the network must be flexible. (3) Offer value cells undergo range
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adaptation but decisions should, in first approximation, not depend on the range of offer values.
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In other words, the network must be adaptable. (4) The activity of neurons in the OFC only
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partially resembles that of neurons in area LIP, for which the model was designed. Specifically,
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in addition to offer value cells (analogous to MT) and chosen juice cells (analogous to LIP), there
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are chosen value cells, which have no known correspondent in perceptual decisions. Moreover,
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even the resemblance between chosen juice cells and neurons in LIP is weak, because the
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representation of chosen juice cells is categorical while that of LIP is continuous, and because
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chosen juice cells do not present the race-to-threshold and the working memory activity the
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model was originally designed to reproduce. In spite of all these challenges, the spirit of this
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study was not to design a new model. Rather, we tested Wang's model as much as possible off-
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the-shelf, without modifying its structure or even adjusting its parameters. We did, however,
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adapt the network in two important ways. First, we strengthened the connections between OV
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cells and CJ cells (reflecting the fact that offer value cells and chosen juice cells are found in the
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same area). Second, we introduced mechanisms of synaptic plasticity to account for the required
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flexibility and adaptability of the model.
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Several traits make W11 a particularly attractive model. First, the model is biophysically realistic.
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Elements in the extended, spiking network are neurons endowed with realistic synaptic
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connections and time constants, and the mean-field approach preserves the realism of the model.
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Second, the model is nearly without free parameters. In other words, the model is very
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constrained, especially when tested outside its original domain. Third, the model makes testable
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predictions on the excitatory/inhibitory nature of different neuronal populations and on their
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connectivity. These traits set W11 apart from other computational models of economic decisions
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including the drift-diffusion models, mutual inhibition models and the leaky competing
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accumulator model (Bogacz et al. 2006; Hare et al. 2011; Krajbich et al. 2010; Usher and
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McClelland 2001). Furthermore, the explicit inclusion of inhibitory interneurons in W11
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provides an account for chosen value cells, which are not explained in more schematic models
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including the drift-diffusion model and the leaky competing accumulator model. Other studies
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examined Wang's model in the context of value-based decisions. In particular, Behrens and
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colleagues used the simplified W2 version to generate aggregate regressors for the analysis of
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MEG data (Hunt et al. 2012; Jocham et al. 2012). In this respect, we note that the tests ran here
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were significantly more stringent because we matched each node in the network with a specific
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group of neurons, because the original analysis of neuronal data (Padoa-Schioppa and Assad
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2006) tested a large number of variables and not only variables generated from the model, and
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because we reproduced a variety of empirical findings.
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We found that W11 provides a remarkably accurate account for the activity of neurons in the
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OFC. The input node of the model (OV cells) can be identified with offer value cells. The model
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naturally generates binary decisions, and the activity profile of the output node (CJ cells) is fairly
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similar to that of chosen juice cells. Perhaps most surprisingly, the activity of inhibitory
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interneurons (CV cells) is very similar to that of chosen value cells (more on this below). In
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addition, W11 reproduces several phenomena related to the neuronal origins of choice variability,
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namely choice hysteresis, the predictive activity of chosen juice cells and the activity
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overshooting of chosen value cells. We examined what changes in the network are necessary to
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ensure that relative values are arbitrary and to accommodate the fact that offer value cells
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undergo range adaptation. We found that these two requirements are addressed only when
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synaptic plasticity is introduced in the connections between OV cells and CJ cells. We also found
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that W11 falls short of the experimental data in at least two ways. First, the model includes only
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neurons with positive encoding (i.e., higher firing rate for higher values). In contrast, for each of
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the three variables, a substantial fraction of the neurons in OFC presents negative encoding (i.e.,
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higher firing rate for lower values) (Padoa-Schioppa 2013; 2009). Second, the model can provide
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the flexibility necessary to accommodate any relative value, or it can handle a significant
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baseline activity in the input node (OV cells), but it cannot do both of these things at the same
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time. As discussed below, these limitations appear conceptually surmountable, although more
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theoretical work is necessary in this respect.
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Chosen value signals and interneurons in the OFC
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One of the main results of this study is that interneurons of W11 were found to encode the
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chosen value. This observation was unforeseen and somewhat extraordinary. Interneurons were
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included in the progenitor models of W11 for biological realism and network stability (Amit and
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Brunel 1997) – not to reproduce empirical observations analogous to chosen value cells.
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Furthermore, chosen value cells have no known correspondent in perceptual decision tasks. In
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contrast, chosen value signals have been observed in numerous studies of value-based decisions
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and in multiple brain regions (Amemori and Graybiel 2012; Cai et al. 2011; Cai and Padoa-
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Schioppa 2014; 2012; Grabenhorst et al. 2012; Lau and Glimcher 2008; Lee et al. 2012; Padoa-
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Schioppa and Assad 2006; Roesch et al. 2009; Strait et al. 2014; Sul et al. 2010; Wunderlich et al.
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2010). While it is clear that chosen value signals could inform a variety of mental functions
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including associative learning, visual attention, emotion and others, the possible contributions of
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chosen value signals to economic decisions have remained mysterious. The present results
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suggest that chosen value cells in the OFC may be directly involved in the decision. This
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suggestion is bolstered by the fact that CV cells in the model reproduce the activity overshooting
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seen in chosen value cells. Importantly, the hypothesis that chosen value cells in the OFC
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participate in the decision does not imply that all chosen value signals in the brain are the
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signature of a decision process. Indeed, once computed and explicitly represented by a neuronal
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population, the variable chosen value could be transmitted broadly to other brain regions and
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thus facilitate various mental functions.
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Considering the similarity between CV cells and chosen value cells from a different angle, W11
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makes the strong prediction that all chosen value cells in the decision circuit are interneurons and
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that all chosen juice cells are pyramidal cells. Future experiments will test these predictions.
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Importantly, the present model provides an account for the decision circuit. However, OFC likely
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comprises additional neuronal populations that do not directly intervene in the decision but
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receive input from the decision circuit. Thus testing the predictions of the model will require
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distinguishing between neurons that contribute to the decisions and other neurons within OFC.
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Notably, the hypothesis that the variable chosen value is first computed in OFC and then
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transmitted to other brain regions implies that this variable is passed from the decision circuit to
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pyramidal cells in layers 5 and 6 of OFC, where cortico-cortical projections originate. In
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principle, NS pyramidal cells in W11 (whose activity resembles that of chosen value cells with
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low dynamic range) could serve that purpose, although alternate schemes are also possible.
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In its current form, W11 presents two clear limitations – the fact that the model only includes
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neurons with positive encoding and the challenge posed by the baseline activity of offer value
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cells. Negative encoding is an intriguing phenomenon, partly because it is not observed in
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sensory systems (to our knowledge). More detailed neurophysiology work is necessary to
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establish the excitatory/inhibitory nature of neurons with negative encoding, whether these
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neurons are preferentially located in specific cortical layers, and whether they present specific
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patterns of connectivity. From a modeling perspective, the presence of cells with negative
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encoding can be viewed as a degree of freedom present in the empirical data but not used by the
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model. In principle, this degree of freedom could help resolve the challenge posed by the
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baseline activity in offer value cells, because two inputs encoding the same variable with
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opposite sign (excitatory and inhibitory) could be combined in a way that reduces or even
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eliminates any baseline. In the light of these considerations, we view W11 as a benchmark and a
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starting point for biophysically realistic models of economic decisions. The next steps are to
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examine empirically the excitatory/inhibitory nature of different groups of cells in OFC, to assess
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possible differences between cortical layers, and to establish the actual connectivity between the
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different groups of neurons (a set of challenging tasks!). The results of these enquiries should
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then inform new and more accurate neuro-computational models.
27
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To conclude, Wang's model provides a biologically credible account for the neuronal
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mechanisms of economic decisions. In the model, decisions are generated by three groups of
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cells whose activity closely resembles the activity of the three groups of neurons previously
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found in the OFC. This close resemblance does not demonstrate that economic decisions take
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place in the OFC, or that the decision mechanisms are based on recurrent excitation and pooled
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inhibition. However, the model does provide an important proof of concepts that economic
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decisions could emerge from the activity of neurons identified in this area. In this framework,
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W11 suggests that chosen value cells are a key component of the decision process. The model
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makes several non-trivial predictions that require further empirical testing. Last but not least,
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Wang's model represents an important bridge between neuroscience and economics, providing a
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platform to investigate the implications of neuronal data for economic theory. We will examine
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some of these implications in future work.
755 756 757
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Table 1. Parameters used in simulations.
Network parameters (same as in W11) NE NI Cext F rext
1,600 400 800 0.15 3.0 sp/s
Total number of pyramidal cells Total number of interneurons Number of external connections per cell Fraction of pyramidal cells associated to each stimulus/good Mean firing rate of external cells
Time constants, synaptic efficacies and noise (same as in W11) τAMPA τNMDA τGABA JAMPA, ext, pyr JAMPA, rec, pyr JNMDA, rec, pyr JGABA, rec, pyr JAMPA, ext, in JAMPA, rec, in JNMDA, rec, in JGABA, rec, in γ ση
2 ms 100 ms 5 ms -0.1123 -0.0027 -0.00091979 0.0215 -0.0842 -0.0022 -0.00083446 0.0180 0.641 0.020 A
Time constant for AMPA-mediated synapses Time constant for NMDA-mediated synapses Time constant for GABA-mediated synapses Efficacy of external synapses, pyramidal cells Efficacy of recurrent AMPA synapses, pyramidal cells Efficacy of recurrent NMDA synapses, pyramidal cells Efficacy of recurrent GABA synapses, pyramidal cells Efficacy of external synapses, interneurons Efficacy of recurrent AMPA synapses, interneurons Efficacy of recurrent NMDA synapses, interneurons Efficacy of recurrent GABA synapses, interneurons Proportionality between input firing rate and gating variable Noise parameter of the Ornstein-Uhlenbeck process
Parameters of input-output function for integrate-and-fire neurons (same as in W11) IE gE cE II gI cI
125 0.16 310 177 0.087 615
Threshold current, pyramidal cells Gain factor, pyramidal cells Noise factor, pyramidal cells Threshold current, interneurons Gain factor, interneurons Noise factor, interneurons
Parameters used to model OV cells r0 Δr a b c d JAMPA, input
0 or 6 sp/s 8 sp/s toffer + 175 ms 30 ms toffer + 400 ms 100 ms 30*JAMPA, ext, pyr
Baseline activity Dynamic range See Eq.23 See Eq.23 See Eq.23 See Eq.23 Efficacy of AMPA-mediated synapses from OV to CJ cells
759 760
29
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Figure legends
762
Figure 1. Summary of experimental results. a. Neuronal response encoding the offer value B. In
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the left panel, the x-axis represents different offer types. Black dots represent the percent of trials
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in which the animal chose juice B and red symbols indicate the neuronal firing rate. Each symbol
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represents one trial type. Diamonds and circle indicate trials in which the animal chose juice A
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and juice B, respectively. In the right panel, the same neuronal response (y-axis) is plotted
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against the encoded variable (x-axis). It can be noted that the encoding is close to linear. b.
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Activity profile for offer value cells. The plot illustrates the population activity. For each neuron,
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trials were divided in three groups (tertiles) based on the value of the encoded juice (low,
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medium, high). A trace was computed for each group of by averaging the activity across trials,
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and the resulting traces were averaged across the population. Only cells with positive encoding
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are shown here. c. Neuronal response encoding the chosen juice. Same conventions as in panel a.
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In this case, the activity is roughly binary depending on the juice chosen by the animal. d.
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Activity profile for chosen juice cells. In this case, trials were divided depending on whether the
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animal chose the juice encoded by the cell (E chosen) or the other juice (O chosen). Here the
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"encoded" juice was defined as that which elicited higher activity. Traces are population
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averages. e. Neuronal response encoding the chosen value. Same conventions as in panel a. f.
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Activity profile for chosen value cells. In this case, trials were divided in three groups (tertiles)
779
depending on the chosen value (low, medium, high). Traces are population averages. The figure
780
is adapted from (Padoa-Schioppa 2013).
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Figure 2. Schematics of W11. Excitatory and inhibitory connections are indicated in blue and
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red, respectively. Different arrow heads indicate synaptic efficacies between different groups of
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cells (see figure legend and corresponding numerical values in Table 1).
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Figure 3. Illustration of inputs for random dot task (left) and economic choice task (right).
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Figure 4. Initial test of Wang's model. a. OVB cells, activity profile. Trials were divided in three
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groups (tertiles) depending on the offer value B (same convention as in Fig.1b). b. OVB cells,
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tuning curve. The firing rate (z-axis) is plotted as a function of the two offers (x- and y-axes).
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Each data point represents one trial type. Red diamonds and blue circles represent trial types in
30
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which the network chose juice A and juice B, respectively. c. OVB cells, reduced tuning curve.
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Down-sampled version of the data shown in b, displayed in the format of Fig.1a. d. OVB cells,
791
activity versus encoded variable. Same firing rates shown in panel b plotted as a function of the
792
encoded variable (offer value B). efgh. Activity of CJB cells. In panel e, trials were divided in
793
two groups depending on the chosen juice (same convention as in Fig.1d). ijkl. activity of CV
794
cells. In panel i, trials were divided in three groups (tertiles) depending on the chosen value
795
(same conventions as in Fig.1f). In all panels, red diamonds (blue circles) represent choices of
796
juice A (juice B).
797
Figure 5. Choice patterns obtained for different balances of synaptic efficacies. a. Choice pattern
798
for the simulation illustrated in Fig.4. Each data point (gray dots) represents one offer type. The
799
percent of B choices (z-axis) is plotted as a function of the two offers (x- and y-axes). The
800
surface illustrates the result of a logistic fit including all second order terms (Eq.1). b. Same
801
surface as in panel a viewed from the z-axis. The indifference function, roughly corresponding to
802
the green pixels (see color legend) is a straight line through the origin. In this simulation, δJstim =
803
(2, 1) and all other synaptic weights were balanced. c. Choice pattern obtained with an
804
imbalanced NMDA-mediated reverberation. d. Choice pattern obtained with an imbalanced
805
inhibition. In both panels c and d, parameters were chosen such that the model was roughly
806
indifferent between 10A and 20B. In both cases, the indifference function did not cross the origin.
807
Figure 6. Robustness with respect to the strength of recurring synapses. abcd. Summary of
808
results obtained with w+ = 1.55. Panels a and b summarize the results obtained for CJB cells
809
(same format as in Fig.4ef). Panels c and d summarize the results obtained for CV cells (same
810
format as in Fig.4il). efgh. Summary of results obtained with w+ = 1.70. ijkl. Summary of results
811
obtained with w+ = 1.85.
812
Figure 7. Predictive activity and choice hysteresis. a. Choice hysteresis (experimental data). In
813
this representative session, trials were divided in two groups depending on whether the outcome
814
of the previous trial was juice A (A• trials) or juice B (B• trials). It can be observed that the
815
relative value of juice A was higher in A• trials (dark gray) compared to B• trials (light gray). In
816
other words, all other things equal, the animal had a tendency to choose on any given trial the
817
same juice as in the previous trial. b. Predictive activity of chosen juice cells (experimental data). 31
818
See main text. c. Predictive activity of CJ cells (model). Same simulation as in Fig.4. The
819
activity immediately before and after the offer is enlarged in the insert. It can be noted that the
820
predictive activity is significant but relatively modest compared to that measured experimentally
821
(see main text). d. Lack of choice hysteresis in initial simulations. Same simulation as in panel c
822
and Fig.4. In this simulation, the initial conditions were reset at the beginning of each trial. In the
823
analysis, trials were divided in A• trials and B• trials depending on the previous outcome, and the
824
two data sets were separately fitted with a logistic function. The two fits are completely
825
overlapping, indicating the lack of choice hysteresis. In other words, the predictive activity in
826
panel c corresponds to the "residual" predictive activity measured in OFC (see main text). ef.
827
Predictive activity in the presence of choice hysteresis. In this simulation, we set the initial
828
conditions in each trial equal to the final conditions in the previous trial (tail activity) and w+ =
829
1.82. To ensure that each juice was chosen in ~50% of trials, we set in this simulation ΔA = [0,
830
10] and ΔB = [0, 20]. Under these conditions, the predictive activity is much larger (compare
831
panels e and c) and there is a robust choice hysteresis (panel f). In panel f, the two logistic fit
832
refer to A• trials and B• trials, respectively. Panels a and b are reproduced from (Padoa-Schioppa
833
2013).
834
Figure 8. Activity overshooting. a. Overshooting of chosen value cells (experimental data). Only
835
trials in which the animal chose 1A are included in the figure. See main text for details. b.
836
Overshooting of CV cells (model). Same simulation as in Fig.4. For each quantity of A that
837
induced some split decisions (i.e., offer value A = 1...10), trials in which the network chose juice
838
A were divided in easy and split (see main text). The activity overshooting was observed for
839
each value of A. The main plot here illustrates the effect averaged across values of A. The insert
840
illustrates the effect for offer value A = 8. Panel a is reproduced from (Padoa-Schioppa 2013).
841
Figure 9. Stable points of W11. The leftmost column (panels a1-a3) represents the space of
842
possible offers (as in Fig.3, right panel). The other three columns represent the results of three
843
simulations, with w+ = 1.75 and ΔJ = 30 (normal parameters, panels b1-b3), w+ = 1.85 and ΔJ =
844
30 (panels c1-c3) and w+ = 1.75 and ΔJ = 15 (panels d1-d3). In each panel, the two axes
845
represent the "final" activity of CJA cells (y-axis) and CJB cells (x-axis) (time window 400-600
846
ms after the offer). Each data point represents one trial and data points are color-coded according
847
to the legend in the corresponding leftmost column (see main text). 32
848
Figure 10. Effects of non-zero baseline in OV cells. a-f. Symmetric network. Panel a shows the
849
simulated activity of OVB cells, panels bc show the activity of CJB cells, panels de show the
850
activity of CV cells and panel f shows the choice pattern. The presence of the baseline activity in
851
OV cells changes dramatically the nature of the input. However, with small adjustments, the
852
symmetric network is robust to this change. In this simulation, we set w+ = 1.65, JAMPA, rec, in = 0.9
853
JAMPA, rec, in, JNMDA, rec, in = 0.9 JNMDA, rec, in, JGABA, rec, in = 0.8 JGABA, rec, in. The activity profiles of CJ
854
and CV cells remain comparable to the empirical observations, and CV cells encode the chosen
855
value. g-l. Asymmetric network. In this simulation, we set δJstim = (1.2, 1) all other parameters as
856
in panels a-f. Although the activity profiles of CJ cells (panels hi) and CV cells (panels j) are
857
realistic, the indifference function is non-homogenous. Tests in which we tried to recover this
858
departure from homogeneity introducing imbalances at other stages of the network did not
859
provide successful results.
860 861
33
862
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37
Figure 1
1A = 3.1B
2
R = 0.92 30
10
10
0
0
0B:1A 1B:3A 1B:2A 1B:1A 2B:1A 3B:1A 4B:1A 6B:1A 10B:1A 2B:0A
20
0%
20
5
2B:0A
6B:1A
4B:1A
3B:1A
2B:1A
0
1B:1A
0 1B:2A
10 0%
10
A
0%
0B:1A 1B:3A 1B:2A 1B:1A 2B:1A 3B:1A 4B:1A 6B:1A 10B:1A 3B:0A
Average firing rate (sp/s)
Firing rate (sp/s)
100%
20
10 0
12
10
20
f
2
20
14
offer
R = 0.90 30
16
8
40
30
E chosen O chosen
B
chosen juice
e Chosen value cells
10
8
18
Average firing rate (sp/s)
40
20
1B:3A
d 2
20
1A = 3.2B
10
offer
R = 0.95 30
40
12
10
50
30
0B:1A
Firing rate (sp/s)
40
14
offer value B
100%
1A = 1.9B
high medium low
6 0
c Chosen juice cells 50
16
b Average firing rate (sp/s)
30
100%
Firing rate (sp/s)
a Offer value cells
5
chosen value
10
high medium low
18 16 14 12 10 8
offer
500 ms
Figure 2
CV chosen value
NS non selective
CJA
CJB
chosen juice A
chosen juice B
OVA
OVB
offer value A
offer value B
Synaptic efficacies JGABA, rec, int
JAMPA, rec, pyr JAMPA, ext, pyr
JNMDA, rec, int
JAMPA, input
JAMPA, rec, int
Time constants
JAMPA, ext, int JGABA, rec, pyr JNMDA, rec, pyr
WGABA WNMDA WAMPA
Figure 3
economic choice task
random dot task 'A
offer A
motion strenght right
0
í í
0 motion strenght left
0
'B
0 offer B
Figure 4
OVB cells high medium low
6
b
c
A chosen B chosen
5
5
4
4
3
3
3
2
2
2
1
1
1
0
0
4 4
d
5
2
0 20
0 -500
0
500
0 10
1000
0
offer A
offer
20
10
0
offer B
0B:1A 1B:20A 1B:16A 1B:12A 1B:8A 1B:4A 4B:1A 8B:1A 12B:1A 16B:1A 20B:1A 1B:0A
Firing rate (sp/s)
a
0
5
10
15
20
offer value B
CJB cells A chosen B chosen
25
f
g
A chosen B chosen
h
15
15
10
10
10
5
5
5
0
0
15 20 15 10 5 0 20
0 -500
0
500
10
1000
0
offer A
offer
20
10
0
0B:1A 1B:20A 1B:16A 1B:12A 1B:8A 1B:4A 4B:1A 8B:1A 12B:1A 16B:1A 20B:1A 1B:0A
Firing rate (sp/s)
e
offer B
A chosen
B chosen
CV cells high medium low
18
j
k
A chosen B chosen
l
16
16
16
14
14
14
12
12
10
10
16
14 12 12
10 -500
10 20 0
offer
500
1000
10
offer A
0
0
5
10
offer B
15
20
0B:1A 1B:20A 1B:16A 1B:12A 1B:8A 1B:4A 4B:1A 8B:1A 12B:1A 16B:1A 20B:1A 1B:0A
Firing rate (sp/s)
i
0
10
20
30
chosen value (uB)
40
Figure 5
a
b
20
100 15
offer A
% B choice
75
50
10
25 5 0 20 20
15 15
10 5
d
0
5
0
0
10
15
20
15
20
offer B
5
offer A
100
0
10
offer B
c
20
20
90 80
15
15
50 40
offer A
60
offer A
70
10
10
30 20
5
5
10 0 0
5
10
offer B
15
20
0
0
5
10
offer B
Figure 6
CJB cells
a
A chosen B chosen
4
b
CV cells
c
A chosen B chosen
Firing rate (sp/s)
w+ = 1.55
3
í
0
14 14
9
4
6
12
12 2 20
3 -500
0
500
1000
10
offer A
0
0
10
20
10 -500
offer B
0
500
10 0
1000
offer
g
f 4
Firing rate (sp/s)
í
20
30
40
h
18
12
16
20 3
10
chosen value (uB)
e w+ = 1.70
A chosen B chosen
16
16
6 12
d
high medium low
15
16
0
8
15
14 14
10
4 12 12
5 0 20 0 -500
0
500
1000
10
offer A
0
0
10
20
offer B
10 -500
0
500
10 0
1000
10
20
30
40
chosen value (uB)
offer
i
j
k
30
l
Firing rate (sp/s)
w+ = 1.85
4
18
30 3
í
16
20
0
16 20
14 14
10 10
12 12 0 20
0 -500
0
offer
500
1000
10
offer A
0
0
10
offer B
20
10 -500
0
500
1000
10 0
10
20
30
chosen value (uB)
40
Figure 7
a
b
choice hysteresis
$WULDOV B• trials
chosen juice cells E chosen, easy E chosen, split O chosen, split O chosen, easy
% of B choices
Firing rate (sp/s)
14
12
8 % $
% $
% $ % $ % $ % $
% $
% $ % $
% $
í
offer
d
c $WULDOV B• trials
CJB cells
4
Firing rate (sp/s)
%B choice
RIIHU$
f
offer
e
$WULDOV B• trials
í
offer B
8 6
Firing rate (sp/s)
%B choice
3
í í í
RIIHU$
offer B
4 2 í í í
í
offer
Figure 8
a
Trials: A chosen easy split
12
Firing rate (sp/s)
b
chosen value cells
CV cells
15 16
14
14 12 í
0
500
1000
13
10
12 8 11 -500
0
offer
500
1000
-500
0
offer
500
1000
Figure 9
w+ = 1.75, 'J = 30 15
20
a1
w+ = 1.85, 'J = 30 b1
w+ = 1.75, 'J = 15 c1
15
d1
firing rate CJA cells
30
offer A
10
5
0
0
5
10
10 20 10
0 0
15
5
10
offer B
10
20
0 0
10
20
0 0
30
5
10
15
firing rate CJB cells
15
a2
20
b2
c2
15
d2
30 10
10 20 10 5
0
5
10
0
5
10
0 0
15
a3
15
10
20
20
0 0
10
20
0 0
30
b3
c3
5
10
15
15
d3
30 10
10 20 10 5
0
5
10
0
5
10
15
0 0
10
20
0 0
10
20
30
0 0
5
10
15
Figure 10
a
20
high medium low
12
b
CJB cells
8 10 6 4
5
0
500
0 í500
1000
offer
0
500
f
0
2 20
10
20
offer B
d
high medium low
12
0
12
20
10
0
offer B
e
12
high medium low
A chosen B chosen
10 10
8 í500
0
500
8 0
1000
5
offer
g
OVB cells
10
offer A
CVcells Firing rate (sp/s)
offer A
10
0
1000
offer
20
Firing rate (sp/s)
A chosen B chosen 10
15
0 500
10
15
20
chosen value (uB)
h
CJB cells 20
A chosen B chosen
i A chosen B chosen
9
15 6 8 10 3 4
5
0 500
0
500
1000
0 í500
offer
0
500
1000
offer
l 20
offer A
c
A chosen B chosen
0 20
10
offer A
j
CVcells 14
high medium low
0
0
20
10
offer B
k
12 A chosen B chosen
12
11
10
10
10
0
0
10
offer B
20
8 500
0
offer
500
1000
9 0B:1A 1B:20A 1B:16A 1B:12A 1B:8A 1B:4A 4B:1A 8B:1A 12B:1A 16B:1A 20B:1A 1B:0A
Firing rate (sp/s)
OVB cells
