Consideration of the typical workflow of theoretical modeling clarifies the need for emergent property inference. First, one designs or chooses an existing circuit model that, it is hypothesized, captures the computation of interest. To ground this process in a well-known example, consider the stomatogastric ganglion (STG) of crustaceans, a small neural circuit which generates multiple rhythmic muscle activation patterns for digestion (Marder and Thirumalai, 2002). Despite full knowledge of STG connectivity and a precise characterization of its rhythmic pattern generation, biophysical models of the STG have complicated relationships between circuit parameters and computation (Goldman et al., 2001; Prinz et al., 2004).

A subcircuit model of the STG (Gutierrez et al., 2013) is shown schematically in Figure 1A. The fast population (f1 and f2) represents the subnetwork generating the pyloric rhythm and the slow population (s1 and s2) represents the subnetwork of the gastric mill rhythm. The two fast neurons mutually inhibit one another, and spike at a greater frequency than the mutually inhibiting slow neurons. The hub neuron couples with either the fast or slow population, or both depending on modulatory conditions. The jagged connections indicate electrical coupling having electrical conductance $g_{\text{el}}$, smooth connections in the diagram are inhibitory synaptic projections having strength $g_{\text{synA}}$ onto the hub neuron, and $g_{\text{synB}}=5$ nS for mutual inhibitory connections. Note that the behavior of this model will be critically dependent on its parameterization – the choices of conductance parameters $\mathbf{𝐳}=[g_{\text{el}},g_{\text{synA}}]$.

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Second, once the model is selected, one must specify what the model should produce. In this STG model, we are concerned with neural spiking frequency, which emerges from the dynamics of the circuit model (Figure 1B). An emergent property studied by Gutierrez et al. is the hub neuron firing at an intermediate frequency between the intrinsic spiking rates of the fast and slow populations. This emergent property (EP) is shown in Figure 1C at an average frequency of 0.55 Hz. To be precise, we define intermediate hub frequency not strictly as 0.55 Hz, but frequencies of moderate deviation from 0.55 Hz between the fast (.35Hz) and slow (.68Hz) frequencies.

Third, the model parameters producing the emergent property are inferred. By precisely quantifying the emergent property of interest as a statistical feature of the model, we use emergent property inference (EPI) to condition directly on this emergent property. Before presenting technical details (in the following section), let us understand emergent property inference schematically. EPI (Figure 1D) takes, as input, the model and the specified emergent property, and as its output, returns the parameter distribution (Figure 1E). This distribution – represented for clarity as samples from the distribution – is a parameter distribution constrained such that the circuit model produces the emergent property. Once EPI is run, the returned distribution can be used to efficiently generate additional parameter samples. Most importantly, the inferred distribution can be efficiently queried to quantify the parametric structure that it captures. By quantifying the parametric structure governing the emergent property, EPI informs the central question of this inverse problem: what aspects or combinations of model parameters have the desired emergent property?

EPI formalizes the three-step procedure of the previous section with deep probability distributions (Rezende and Mohamed, 2015; Papamakarios et al., 2019a). First, as is typical, we consider the model as a coupled set of noisy differential equations. In this STG example, the model activity (or state) $\mathbf{𝐱}=[x_{\text{f1}},x_{\text{f2}},x_{\text{hub}},x_{\text{s1}},x_{\text{s2}}]$ is the membrane potential for each neuron, which evolves according to the biophysical conductance-based equation:

where $C_m$ = 1nF, and $\mathbf{𝐡}$ is a sum of the leak, calcium, potassium, hyperpolarization, electrical, and synaptic currents, all of which have their own complicated dependence on activity $\mathbf{𝐱}$ and parameters $\mathbf{𝐳}=[g_{\text{el}},g_{\text{synA}}]$, and $d\mathbf{𝐁}$ is white gaussian noise (Gutierrez et al., 2013; see Section 'STG model' for more detail).

Second, we determine that our model should produce the emergent property of ‘intermediate hub frequency’ (Figure 1C). We stipulate that the hub neuron’s spiking frequency – denoted by statistic ${\omega }_{\text{hub}}(\mathbf{𝐱})$ – is close to a frequency of 0.55 Hz, between that of the slow and fast frequencies. Mathematically, we define this emergent property with two constraints: that the mean hub frequency is 0.55 Hz,

and that the variance of the hub frequency is moderate

In the emergent property of intermediate hub frequency, the statistic of hub neuron frequency is an expectation over the distribution of parameters $\mathbf{𝐳}$ and the distribution of the data $\mathbf{𝐱}$ that those parameters produce. We define the emergent property $𝒳$ as the collection of these two constraints. In general, an emergent property is a collection of constraints on statistical moments that together define the computation of interest.

Third, we perform emergent property inference: we find a distribution over parameter configurations $\mathbf{𝐳}$ of models that produce the emergent property; in other words, they satisfy the constraints introduced in Equations 2 and 3. This distribution will be chosen from a family of probability distributions $𝒬=\{q_{\mathit{𝜽}}(\mathbf{𝐳}):\mathit{𝜽}\in \mathrm{\Theta }\}$, defined by a deep neural network (Rezende and Mohamed, 2015; Papamakarios et al., 2019a; Figure 1D, EPI box). Deep probability distributions map a simple random variable ${\mathbf{𝐳}}_0$ (e.g. an isotropic gaussian) through a deep neural network with weights and biases $\mathit{𝜽}$ to parameters $\mathbf{𝐳}=g_{\mathit{𝜽}}({\mathbf{𝐳}}_0)$ of a suitably complicated distribution (see Section 'Deep probability distributions and normalizing flows' for more details). Many distributions in $𝒬$ will respect the emergent property constraints, so we select the most random (highest entropy) distribution, which also means this approach is equivalent to bayesian variational inference (see Section 'EPI as variational inference'). In EPI optimization, stochastic gradient steps in $\mathit{𝜽}$ are taken such that entropy is maximized, and the emergent property $𝒳$ is produced (see Section 'Emergent property inference (EPI)'). We then denote the inferred EPI distribution as $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳)$, since the structure of the learned parameter distribution is determined by weights and biases $\mathit{𝜽}$, and this distribution is conditioned upon emergent property $𝒳$.

The structure of the inferred parameter distributions of EPI can be analyzed to reveal key information about how the circuit model produces the emergent property. As probability in the EPI distribution decreases away from the mode of $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳)$ (Figure 1E yellow star), the emergent property deteriorates. Perturbing $\mathbf{𝐳}$ along a dimension in which $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳)$ changes little will not disturb the emergent property, making this parameter combination robust with respect to the emergent property. In contrast, if $\mathbf{𝐳}$ is perturbed along a dimension with strongly decreasing $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳)$, that parameter combination is deemed sensitive (Raue et al., 2009; Raman et al., 2017). By querying the second-order derivative (Hessian) of $\log q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳)$ at a mode, we can quantitatively identify how sensitive (or robust) each eigenvector is by its eigenvalue; the more negative, the more sensitive and the closer to zero, the more robust (see Section 'Hessian sensitivity vectors'). Indeed, samples equidistant from the mode along these dimensions of sensitivity (${\mathbf{𝐯}}_1$, smaller eigenvalue) and robustness (${\mathbf{𝐯}}_2$, greater eigenvalue) (Figure 1E, arrows) agree with error contours (Figure 1E contours) and have diminished or preserved hub frequency, respectively (Figure 1F activity traces). The directionality of ${\mathbf{𝐯}}_2$ suggests that changes in conductance along this parameter combination will most preserve hub neuron firing between the intrinsic rates of the pyloric and gastric mill rhythms. Importantly and unlike alternative techniques, once an EPI distribution has been learned, the modes and Hessians of the distribution can be measured with trivial computation (see Section 'Deep probability distributions and normalizing flows').

In the following sections, we demonstrate EPI on three neural circuit models across ranges of biological realism, neural system function, and network scale. First, we demonstrate the superior scalability of EPI compared to alternative techniques by inferring high-dimensional distributions of recurrent neural network connectivities that exhibit amplified, yet stable responses. Next, in a model of primary visual cortex (Litwin-Kumar et al., 2016; Palmigiano et al., 2020), we show how EPI discovers parametric degeneracy, revealing how input variability across neuron types affects the excitatory population. Finally, in a model of superior colliculus (Duan et al., 2021), we used EPI to capture multiple parametric regimes of task switching, and queried the dimensions of parameter sensitivity to characterize each regime.

To understand how EPI scales in comparison to existing techniques, we consider recurrent neural networks (RNNs). Transient amplification is a hallmark of neural activity throughout cortex and is often thought to be intrinsically generated by recurrent connectivity in the responding cortical area (Murphy and Miller, 2009; Hennequin et al., 2014; Bondanelli et al., 2019). It has been shown that to generate such amplified, yet stabilized responses, the connectivity of RNNs must be non-normal (Goldman, 2009; Murphy and Miller, 2009), and satisfy additional constraints (Bondanelli and Ostojic, 2020). In theoretical neuroscience, RNNs are optimized and then examined to show how dynamical systems could execute a given computation (Sussillo, 2014; Barak, 2017), but such biologically realistic constraints on connectivity (Goldman, 2009; Murphy and Miller, 2009; Bondanelli and Ostojic, 2020) are ignored for simplicity or because constrained optimization is difficult. In general, access to distributions of connectivity that produce theoretical criteria like stable amplification, chaotic fluctuations (Sompolinsky et al., 1988), or low tangling (Russo et al., 2018) would add scientific value to existing research with RNNs. Here, we use EPI to learn RNN connectivities producing stable amplification, and demonstrate the superior scalability and efficiency of EPI to alternative approaches.

We consider a rank-2 RNN with $N$ neurons having connectivity $W=U{V}^{\top }$ and dynamics

where $U=\left[\begin{array}{cc}\hfill {\mathbf{𝐔}}_1\hfill & \hfill {\mathbf{𝐔}}_2\hfill \end{array}\right]+g{\chi }^{(U)}$, $V=\left[\begin{array}{cc}\hfill {\mathbf{𝐕}}_1\hfill & \hfill {\mathbf{𝐕}}_2\hfill \end{array}\right]+g{\chi }^{(V)}$, ${\mathbf{𝐔}}_1{\mathbf{𝐔}}_2,{\mathbf{𝐕}}_1,{\mathbf{𝐕}}_2\in {[-1,1]}^N$, and ${\chi }_{i,j}^{(U)},{\chi }_{i,j}^{(V)}\sim 𝒩(0,1)$. We infer connectivity parameters $\mathbf{𝐳}=[{\mathbf{𝐔}}_1,{\mathbf{𝐔}}_2,{\mathbf{𝐕}}_1,{\mathbf{𝐕}}_2]$ that produce stable amplification. Two conditions are necessary and sufficient for RNNs to exhibit stable amplification (Bondanelli and Ostojic, 2020): $\text{real}({\lambda }_1)<1$ and ${\lambda }_1^s>1$, where ${\lambda }_1$ is the eigenvalue of $W$ with greatest real part and ${\lambda }^s$ is the maximum eigenvalue of $W^s=\frac{W+W^{\top }}{2}$. RNNs with $\text{real}({\lambda }_1)=0.5\pm 0.5$ and ${\lambda }_1^s=1.5\pm 0.5$ will be stable with modest decay rate ($\text{real}({\lambda }_1)$ close to its upper bound of 1) and exhibit modest amplification (${\lambda }_1^s$ close to its lower bound of 1). EPI can naturally condition on this emergent property

Variance constraints predicate that the majority of the distribution (within two standard deviations) are within the specified ranges.

For comparison, we infer the parameters $\mathbf{𝐳}$ likely to produce stable amplification using two alternative simulation-based inference approaches. Sequential Monte Carlo approximate bayesian computation (SMC-ABC) (Sisson et al., 2007) is a rejection sampling approach that uses SMC techniques to improve efficiency, and sequential neural posterior estimation (SNPE) (Gonçalves et al., 2019) approximates posteriors with deep probability distributions (see Section 'Related approaches'). Unlike EPI, these statistical inference techniques do not constrain the predictions of the inferred distribution, so they were run by conditioning on an exemplar dataset ${\mathbf{𝐱}}_0=\mathit{𝝁}$, following standard practice with these methods (Sisson et al., 2007; Gonçalves et al., 2019). To compare the efficiency of these different techniques, we measured the time and number of simulations necessary for the distance of the predictive mean to be less than 0.5 from $\mathit{𝝁}={\mathbf{𝐱}}_0$ (see Section 'Scaling EPI for stable amplification in RNNs').

As the number of neurons $N$ in the RNN, and thus the dimension of the parameter space $\mathbf{𝐳}\in {[-1,1]}^{4N}$, is scaled, we see that EPI converges at greater speed and at greater dimension than SMC-ABC and SNPE (Figure 2A). It also becomes most efficient to use EPI in terms of simulation count at $N=50$ (Figure 2B). It is well known that ABC techniques struggle in parameter spaces of modest dimension (Sisson et al., 2018), yet we were careful to assess the scalability of SNPE, which is a more closely related methodology to EPI. Between EPI and SNPE, we closely controlled the number of parameters in deep probability distributions by dimensionality (Figure 2—figure supplement 1), and tested more aggressive SNPE hyperparameter choices when SNPE failed to converge (Figure 2—figure supplement 2). In this analysis, we see that deep inference techniques EPI and SNPE are far more amenable to inference of high dimensional RNN connectivities than rejection sampling techniques like SMC-ABC, and that EPI outperforms SNPE in both wall time (elapsed real time) and simulation count.

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No matter the number of neurons, EPI always produces connectivity distributions with mean and variance of $\text{real}({\lambda }_1)$ and ${\lambda }_1^s$ according to $𝒳$ (Figure 2C, blue). For the dimensionalities in which SMC-ABC is tractable, the inferred parameters are concentrated and offset from the exemplar dataset ${\mathbf{𝐱}}_0$ (Figure 2C, green). When using SNPE, the predictions of the inferred parameters are highly concentrated at some RNN sizes and widely varied in others (Figure 2C, orange). We see these properties reflected in simulations from the inferred distributions: EPI produces a consistent variety of stable, amplified activity norms $|\mathbf{𝐱}(t)|$, SMC-ABC produces a limited variety of responses, and the changing variety of responses from SNPE emphasizes the control of EPI on parameter predictions (Figure 2D). Even for moderate neuron counts, the predictions of the inferred distribution of SNPE are highly dependent on $N$ and $g$, while EPI maintains the emergent property across choices of RNN (see Section 'Effect of RNN parameters on EPI and SNPE inferred distributions').

To understand these differences, note that EPI outperforms SNPE in high dimensions by using gradient information (from ${\nabla }_{\mathbf{𝐳}}{[\text{real}({\lambda }_1),{\lambda }_1^s]}^{\top }$). This choice agrees with recent speculation that such gradient information could improve the efficiency of simulation-based inference techniques (Cranmer et al., 2020), as well as reflecting the classic tradeoff between gradient-based and sampling-based estimators (scaling and speed versus generality). Since gradients of the emergent property are necessary in EPI optimization, gradient tractability is a key criteria when determining the suitability of a simulation-based inference technique. If the emergent property gradient is efficiently calculated, EPI is a clear choice for inferring high dimensional parameter distributions. In the next two sections, we use EPI for novel scientific insight by examining the structure of inferred distributions.

Dynamical models of excitatory (E) and inhibitory (I) populations with supralinear input-output function have succeeded in explaining a host of experimentally documented phenomena in primary visual cortex (V1). In a regime characterized by inhibitory stabilization of strong recurrent excitation, these models give rise to paradoxical responses (Tsodyks et al., 1997), selective amplification (Goldman, 2009; Murphy and Miller, 2009), surround suppression (Ozeki et al., 2009), and normalization (Rubin et al., 2015). Recent theoretical work (Hennequin et al., 2018) shows that stabilized E-I models reproduce the effect of variability suppression (Churchland et al., 2010). Furthermore, experimental evidence shows that inhibition is composed of distinct elements – parvalbumin (P), somatostatin (S), VIP (V) – composing 80% of GABAergic interneurons in V1 (Markram et al., 2004; Rudy et al., 2011; Tremblay et al., 2016), and that these inhibitory cell types follow specific connectivity patterns (Figure 3A; Pfeffer et al., 2013). Here, we use EPI on a model of V1 with biologically realistic connectivity to show how the structure of input across neuron types affects the variability of the excitatory population – the population largely responsible for projecting to other brain areas (Felleman and Van Essen, 1991).

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We considered response variability of a nonlinear dynamical V1 circuit model (Figure 3A) with a state comprised of each neuron-type population’s rate $\mathbf{𝐱}={[x_E,x_P,x_S,x_V]}^{\top }$. Each population receives recurrent input $W\mathbf{𝐱}$, where $W$ is the effective connectivity matrix (see Section 'Primary visual cortex') and an external input with mean $\mathbf{𝐡}$, which determines population rate via supralinear nonlinearity ${\displaystyle \varphi (\cdot )={[\cdot ]}_{+}^2}$. The external input has an additive noisy component $\mathit{ϵ}$ with variance ${\mathit{𝝈}}^2=[{\sigma }_E^2,{\sigma }_P^2,{\sigma }_S^2,{\sigma }_V^2]$. This noise has a slower dynamical timescale ${\tau }_{\text{noise}}>\tau$ than the population rate, allowing fluctuations around a stimulus-dependent steady-state (Figure 3B). This model is the stochastic stabilized supralinear network (SSSN) (Hennequin et al., 2018)

generalized to have multiple inhibitory neuron types. It introduces stochasticity to four neuron-type models of V1 (Litwin-Kumar et al., 2016). Stochasticity and inhibitory multiplicity introduce substantial complexity to the mathematical treatment of this problem (see Section 'Primary visual cortex: Mathematical intuition and challenges') motivating the analysis of this model with EPI. Here, we consider fixed weights $W$ and input $\mathbf{𝐡}$ (Palmigiano et al., 2020), and study the effect of input variability $\mathbf{𝐳}={[{\sigma }_E,{\sigma }_P,{\sigma }_S,{\sigma }_V]}^{\top }$ on excitatory variability.

We quantify levels of E-population variability by studying two emergent properties

where ${\displaystyle s_E(\mathbf{x};\mathbf{z})}$ is the standard deviation of the stochastic ${\displaystyle E}$-population response about its steady state (Figure 3C). In the following analyses, we select 1 Hz² variance such that the two emergent properties do not overlap in $s_E(\mathbf{𝐳};\mathbf{𝐱})$.

First, we ran EPI to obtain parameter distribution $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳(5\text{Hz}))$ producing E-population variability around 5 Hz (Figure 3D). From the marginal distribution of ${\sigma }_E$ and ${\sigma }_P$ (Figure 3D, top-left), we can see that $s_E(\mathbf{𝐱};\mathbf{𝐳})$ is sensitive to various combinations of ${\sigma }_E$ and ${\sigma }_P$. Alternatively, both ${\sigma }_S$ and ${\sigma }_V$ are degenerate with respect to $s_E(\mathbf{𝐱};\mathbf{𝐳})$ evidenced by the unexpectedly high variability in those dimensions (Figure 3D, bottom-right). Together, these observations imply a curved path with respect to $s_E(\mathbf{𝐱};\mathbf{𝐳})$ of 5 Hz, which is indicated by the modes along ${\sigma }_P$ (Figure 3E).

Figure 3E suggests a quadratic relationship in E-population fluctuations and the standard deviation of E- and P-population input; as the square of either ${\sigma }_E$ or ${\sigma }_P$ increases, the other compensates by decreasing to preserve the level of $s_E(\mathbf{𝐱};\mathbf{𝐳})$. This quadratic relationship is preserved at greater level of E-population variability $𝒳(10\text{Hz})$ (Figure 3E and Figure 3—figure supplement 1). Indeed, the sum of squares of ${\sigma }_E$ and ${\sigma }_P$ is larger in $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳(10\text{Hz}))$ than $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳(5\text{Hz}))$ (Figure 3F, $p<1\times {10}^{-10}$), while the sum of squares of ${\sigma }_S$ and ${\sigma }_V$ are not significantly different in the two EPI distributions (Figure 3—figure supplement 3, $p=.40$), in which parameters were bounded from 0 to 0.5. The strong interaction between E- and P-population input variability on excitatory variability is intriguing, since this circuit exhibits a paradoxical effect in the P-population (and no other inhibitory types) (Figure 3—figure supplement 4), meaning that the E-population is P-stabilized. Future research may uncover a link between the population of network stabilization and compensatory interactions governing excitatory variability.

EPI revealed the quadratic dependence of excitatory variability on input variability to the E- and P-populations, as well as its independence to input from the other two inhibitory populations. In a simplified model ($\tau ={\tau }_{\text{noise}}$), it can be shown that surfaces of equal variance are ellipsoids as a function of $\mathit{𝝈}$ (see Section 'Primary visual cortex: Mathematical intuition and challenges'). Nevertheless, the sensitive and degenerate parameters are intractable to predict mathematically, since the covariance matrix depends on the steady-state solution of the network (Hennequin et al., 2018; Gardiner, 2009), and terms in the covariance expression increase quadratically with each additional neuron-type population (see also Section 'Primary visual cortex: Mathematical intuition and challenges'). By pointing out this mathematical complexity, we emphasize the value of EPI for gaining understanding about theoretical models when mathematical analysis becomes onerous or impractical.

It has been shown that rats can learn to switch from one behavioral task to the next on randomly interleaved trials (Duan et al., 2015), and an important question is what neural mechanisms produce this computation. In this experimental setup, rats were given an explicit task cue on each trial, either Pro or Anti. After a delay period, rats were shown a stimulus, and made a context (task) dependent response (Figure 4A). In the Pro task, rats were required to orient toward the stimulus, while in the Anti task, rats were required to orient away from the stimulus. Pharmacological inactivation of the SC impaired rat performance, and time-specific optogenetic inactivation revealed a crucial role for the SC on the cognitively demanding Anti trials (Duan et al., 2021). These results motivated a nonlinear dynamical model of the SC containing four functionally defined neuron-type populations. In Duan et al., 2021, a computationally intensive procedure was used to obtain a set of 373 connectivity parameters that qualitatively reproduced these optogenetic inactivation results. To build upon the insights of this previous work, we use the probabilistic tools afforded by EPI to identify and characterize two linked, yet distinct regimes of rapid task switching connectivity.

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In this SC model, there are Pro- and Anti-populations in each hemisphere (left (L) and right (R)) with activity variables $\mathbf{𝐱}={[x_{LP},x_{LA},x_{RP},x_{RA}]}^{\top }$ (Duan et al., 2021). The connectivity of these populations is parameterized by self $sW$, vertical $vW$, diagonal $dW$ and horizontal $hW$ connections (Figure 4B). The input $\mathbf{𝐡}$ is comprised of a positive cue-dependent signal to the Pro- or Anti-populations, a positive stimulus-dependent input to either the Left or Right populations, and a choice-period input to the entire network (see Section 'SC model'). Model responses are bounded from 0 to 1 as a function $\varphi$ of an internal variable $\mathbf{𝐮}$

The model responds to the side with greater Pro neuron activation; for example the response is left if $x_{LP}>x_{RP}$ at the end of the trial. Here, we use EPI to determine the network connectivity $\mathbf{𝐳}={[sW,vW,dW,hW]}^{\top }$ that produces rapid task switching.

Rapid task switching is formalized mathematically as an emergent property with two statistics: accuracy in the Pro task $p_P(\mathbf{𝐱};\mathbf{𝐳})$ and Anti task $p_A(\mathbf{𝐱};\mathbf{𝐳})$. We stipulate that accuracy be on average 0.75 in each task with variance ${.075}^2$

Seventy-five percent accuracy is a realistic level of performance in each task, and with the chosen variance, inferred models will not exhibit fully random responses (50%), nor perfect performance (100%).

The EPI inferred distribution (Figure 4C) produces Pro- and Anti-task accuracies (Figure 4C, bottom-left) consistent with rapid task switching (Equation 9). This parameter distribution has rich structure that is not captured well by simple linear correlations (Figure 4—figure supplement 1). Specifically, the shape of the EPI distribution is sharply bent, matching ground truth structure indicated by brute-force sampling (Figure 4—figure supplement 5). This is most saliently observed in the marginal distribution of $sW$-$hW$ (Figure 4C top-right), where anticorrelation between $sW$ and $hW$ switches to correlation with decreasing $sW$. By identifying the modes of the EPI distribution ${\mathbf{𝐳}}^{*}(sW)$ at different values of $sW$ (Figure 4C red/purple dots), we can quantify this change in distributional structure with the sensitivity dimension ${\mathbf{𝐯}}_1(\mathbf{𝐳})$ (Figure 4C red/purple arrows). Note that the directionality of these sensitivity dimensions at ${\mathbf{𝐳}}^{*}(sW)$ changes distinctly with $sW$, and are perpendicular to the robust dimensions of the EPI distribution that preserve rapid task switching. These two directionalities of sensitivity motivate the distinction of connectivity into two regimes, which produce different types of responses in the Pro and Anti tasks (Figure 4—figure supplement 2).

When perturbing connectivity along the sensitivity dimension away from the modes

Pro-accuracy monotonically increases in both regimes (Figure 4D, top-left). However, there is a stark difference between regimes in Anti-accuracy. Anti-accuracy falls in either direction of ${\mathbf{𝐯}}_1$ in regime 1, yet monotonically increases along with Pro accuracy in regime 2 (Figure 4D, bottom-left). The sharp change in local structure of the EPI distribution is therefore explained by distinct sensitivities: Anti-accuracy diminishes in only one or both directions of the sensitivity perturbation.

To understand the mechanisms differentiating the two regimes, we can make connectivity perturbations along dimensions that only modify a single eigenvalue of the connectivity matrix. These eigenvalues ${\lambda }_{\text{all}}$, ${\lambda }_{\text{side}}$, ${\lambda }_{\text{task}}$, and ${\lambda }_{\text{diag}}$ correspond to connectivity eigenmodes with intuitive roles in processing in this task (Figure 4—figure supplement 3A). For example, greater ${\lambda }_{\text{task}}$ will strengthen internal representations of task, while greater ${\lambda }_{\text{diag}}$ will amplify dominance of Pro and Anti pairs in opposite hemispheres (Section 'Connectivity eigendecomposition and processing modes'). Unlike the sensitivity dimension, the dimensions ${\mathbf{𝐯}}_a$ that perturb isolated connectivity eigenvalues ${\lambda }_a$ for $a\in \{\text{all},\text{side},\text{task},\text{diag}\}$ are independent of ${\mathbf{𝐳}}^{*}(sW)$ (see Section 'Connectivity eigendecomposition and processing modes'), e.g.

Connectivity perturbation analyses reveal that decreasing ${\lambda }_{\text{task}}$ has a very similar effect on Anti accuracy as perturbations along the sensitivity dimension (Figure 4D, middle). The similar effects of perturbations along the sensitivity dimension ${\mathbf{𝐯}}_1({\mathbf{𝐳}}^{*})$ and reduction of task eigenvalue (via perturbations along $-{\mathbf{𝐯}}_{\text{task}}$) suggest that there is a carefully tuned strength of task representation in connectivity regime 1, which if disturbed results in random Anti-trial responses. Finally, we recognize that increasing ${\lambda }_{\text{diag}}$ has opposite effects on Anti-accuracy in each regime (Figure 4D, right). In the next section, we build on these mechanistic characterizations of each regime by examining their resilience to optogenetic inactivation.

During the delay period of this task, the circuit must prepare to execute the correct task according to the presented cue. The circuit must then maintain a representation of task throughout the delay period, which is important for correct execution of the Anti-task. Duan et al. found that bilateral optogenetic inactivation of SC during the delay period consistently decreased performance in the Anti-task, but had no effect on the Pro-task (Figure 5A; Duan et al., 2021). The distribution of connectivities inferred by EPI exhibited this same effect in simulation at high optogenetic strengths γ, which reduce the network activities $\mathbf{𝐱}(t)$ by a factor $1-\gamma$ (Figure 5B) (see Section 'Modeling optogenetic silencing').

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To examine how connectivity affects response to delay period inactivation, we grouped connectivities of the EPI distribution along the continuum linking regimes 1 and 2 of Section 'EPI identifies two regimes of rapid task switching'. $Z(sW)$ is the set of EPI samples for which the closest mode was ${\mathbf{𝐳}}^{*}(sW)$ (see Section 'Mode identification with EPI'). In the following analyses, we examine how error, and the influence of connectivity eigenvalue on Anti-error change along this continuum of connectivities. Obtaining the parameter samples for these analysis with the learned EPI distribution was more than 20,000 times faster than a brute force approach (see Section 'Sample grouping by mode').

The mean increase in Anti-error of the EPI distribution is closest to the experimentally measured value of 7% at $\gamma =0.675$ (Figure 5B, black dot). At this level of optogenetic strength, regime 1 exhibits an increase in Anti-error with delay period silencing (Figure 5C, left), while regime 2 does not. In regime 1, greater ${\lambda }_{\text{task}}$ and ${\lambda }_{\text{diag}}$ decrease Anti-error (Figure 5C, right). In other words, stronger task representations and diagonal amplification make the SC model more resilient to delay period silencing in the Anti-task. This complements the finding from Duan et al., 2021 (Duan et al., 2021) that ${\lambda }_{\text{task}}$ and ${\lambda }_{\text{diag}}$ improve Anti accuracy.

At roughly $\gamma =0.85$ (Figure 5B, gray dot), the Anti-error saturates, while Pro-error remains at zero. Following delay period inactivation at this optogenetic strength, there are strong similarities in the responses of Pro- and Anti-trials during the choice period (Figure 5D, left). We interpreted these similarities to suggest that delay period inactivation at this saturated level flips the internal representation of task (from Anti to Pro) in the circuit model. A flipped task representation would explain why the Anti-error saturates at 50%: the average Anti-accuracy in EPI inferred connectivities is 75%, but average Anti accuracy would be 25% (100% - ${\displaystyle {\mathbb{E}}_{\mathbf{z}}\left[p_P\right]}$) if the internal representation of task is flipped during the delay period.This hypothesis prescribes a model of Anti-accuracy during delay period silencing of $p_{A,\text{opto}}=100\%-p_P$, which is fit closely across both regimes of the EPI inferred connectivities (Figure 5D, right). Similarities between Pro- and Anti-trial responses were not present at the experiment-matching level of $\gamma =0.675$ (Figure 5—figure supplement 2 left) and neither was anticorrelation in $p_P$ and $p_{A,\text{opto}}$ (Figure 5—figure supplement 2 right).

In summary, the connectivity inferred by EPI to perform rapid task switching replicated results from optogenetic silencing experiments. We found that at levels of optogenetic strength matching experimental levels of Anti-error, only one regime actually exhibited the effect. This connectivity regime is less resilient to optogenetic perturbation, and perhaps more biologically realistic. Finally, we characterized the pathology in Anti-error that occurs in both regimes when optogenetic strength is increased to high levels, leading to a mechanistic hypothesis that is experimentally testable. The probabilistic tools afforded by EPI yielded this insight: we identified two regimes and the continuum of connectivities between them by taking gradients of parameter probabilities in the EPI distribution, we identified sensitivity dimensions by measuring the Hessian of the EPI distribution, and we obtained many parameter samples at each step along the continuum at an efficient rate.

All software written for this study is available at https://github.com/cunningham-lab/epi (copy archived at swh:1:rev:38febae7035ca921334a616b0f396b3767bf18d4), Bittner, 2021.

Solving inverse problems is an important part of theoretical neuroscience, since we must understand how neural circuit models and their parameter choices produce computations. Recently, research on machine learning methodology for neuroscience has focused on finding latent structure in large-scale neural datasets, while research in theoretical neuroscience generally focuses on developing precise neural circuit models that can produce computations of interest. By quantifying computation into an emergent property through statistics of the emergent activity of neural circuit models, we can adapt the modern technique of deep probabilistic inference towards solving inverse problems in theoretical neuroscience. Here, we introduce a novel method for statistical inference, which uses deep networks to learn parameter distributions constrained to produce emergent properties of computation.

Consider model parameterization $\mathbf{𝐳}$, which is a collection of scientifically meaningful variables that govern the complex simulation of data $\mathbf{𝐱}$. For example (see Section 'Motivating emergent property inference of theoretical models'), $\mathbf{𝐳}$ may be the electrical conductance parameters of an STG subcircuit, and $\mathbf{𝐱}$ the evolving membrane potentials of the five neurons. In terms of statistical modeling, this circuit model has an intractable likelihood $p(\mathbf{𝐱}\mid \mathbf{𝐳})$, which is predicated by the stochastic differential equations that define the model. From a theoretical perspective, we are less concerned about the likelihood of an exemplar dataset $\mathbf{𝐱}$, but rather the emergent property of intermediate hub frequency (which implies a consistent dataset $\mathbf{𝐱}$).

In this work, emergent properties $𝒳$ are defined through the choice of emergent property statistic $f(\mathbf{𝐱};\mathbf{𝐳})$ (which is a vector of one or more statistics), and its means $\mathit{𝝁}$, and variances ${\mathit{𝝈}}^2$:

In general, an emergent property may be a collection of first-, second-, or higher-order moments of a group of statistics, but this study focuses on the case written in Equation 12. In the STG example, intermediate hub frequency is defined by mean and variance constraints on the statistic of hub neuron frequency ${\omega }_{\text{hub}}(\mathbf{𝐱};\mathbf{𝐳})$ (Equations 2 and 3). Precisely, the emergent property statistics $f(\mathbf{𝐱};\mathbf{𝐳})$ must have means $\mathit{𝝁}$ and variances ${\mathit{𝝈}}^2$ over the EPI distribution of parameters ($\mathbf{𝐳}\sim q_{\mathit{𝜽}}(\mathbf{𝐳})$) and the data produced by those parameters ($\mathbf{𝐱}\sim p(\mathbf{𝐱}\mid \mathbf{𝐳})$), where the inferred parameter distribution $q_{\mathit{𝜽}}(\mathbf{𝐳})$ itself is parameterized by deep network weights and biases $\mathit{𝜽}$.

In EPI, a deep probability distribution $q_{\mathit{𝜽}}(\mathbf{𝐳})$ is optimized to approximate the parameter distribution producing the emergent property $𝒳$. In contrast to simpler classes of distributions like the gaussian or mixture of gaussians, deep probability distributions are far more flexible and capable of fitting rich structure (Rezende and Mohamed, 2015; Papamakarios et al., 2019a). In deep probability distributions, a simple random variable ${\mathbf{𝐳}}_0\sim q_0({\mathbf{𝐳}}_0)$ (we choose an isotropic gaussian) is mapped deterministically via a sequence of deep neural network layers (g₁, . g_l) parameterized by weights and biases $\mathit{𝜽}$ to the support of the distribution of interest:

Such deep probability distributions embed the inferred distribution in a deep network. Once optimized, this deep network representation of a distribution has remarkably useful properties: fast sampling and probability evaluations. Importantly, fast probability evaluations confer fast gradient and Hessian calculations as well.

Given this choice of circuit model and emergent property $𝒳$, $q_{\mathit{𝜽}}(\mathbf{𝐳})$ is optimized via the neural network parameters $\mathit{𝜽}$ to find a maximally entropic distribution $q_{\mathit{𝜽}}^{*}$ within the deep variational family $𝒬=\{q_{\mathit{𝜽}}(\mathbf{𝐳}):\mathit{𝜽}\in \mathrm{\Theta }\}$ that produces the emergent property $𝒳$:

where $H(q_{\mathit{𝜽}}(\mathbf{𝐳}))={𝔼}_{\mathbf{𝐳}}\left[-\log q_{\mathit{𝜽}}(z)\right]$ is entropy. By maximizing the entropy of the inferred distribution $q_{\mathit{𝜽}}$, we select the most random distribution in family $𝒬$ that satisfies the constraints of the emergent property. Since entropy is maximized in Equation 14, EPI is equivalent to bayesian variational inference (see Section 'EPI as variational inference'), which is why we specify the inferred distribution of EPI as conditioned upon emergent property $𝒳$ with the notation $q_{\mathit{𝜽}}(\mathbf{𝐳}\mid 𝒳)$. To run this constrained optimization, we use an augmented lagrangian objective, which is the standard approach for constrained optimization (Bertsekas, 2014), and the approach taken to fit Maximum Entropy Flow Networks (MEFNs) (Loaiza-Ganem et al., 2017). This procedure is detailed in Section 'Augmented lagrangian optimization' and the pseudocode in Algorithm 'Augmented lagrangian optimization'.

In the remainder of Section 'Emergent property inference (EPI)', we will explain the finer details and motivation of the EPI method. First, we explain related approaches and what EPI introduces to this domain (Section 'Related approaches'). Second, we describe the special class of deep probability distributions used in EPI called normalizing flows (Section 'Deep probability distributions and normalizing flows'). Then, we establish the known relationship between maximum entropy distributions and exponential families (Section 'Maximum entropy distributions and exponential families'). Next, we explain the constrained optimization technique used to solve Equation 14 (Section 'Augmented lagrangian optimization'). Then, we demonstrate the details of this optimization in a toy example (Section 'Example: 2D LDS'). Finally, we explain how EPI is equivalent to variational inference (Section 'EPI as variational inference').

In Section 'Motivating emergent property inference of theoretical models' and 'Emergent property inference via deep generative models', we used EPI to infer conductance parameters in a model of the stomatogastric ganglion (STG) (Gutierrez et al., 2013). This five-neuron circuit model represents two subcircuits: that generating the pyloric rhythm (fast population) and that generating the gastric mill rhythm (slow population). The additional neuron (the IC neuron of the STG) receives inhibitory synaptic input from both subcircuits, and can couple to either rhythm dependent on modulatory conditions. There is also a parametric regime in which this neuron fires at an intermediate frequency between that of the fast and slow populations (Gutierrez et al., 2013), which we infer with EPI as a motivational example. This model is not to be confused with an STG subcircuit model of the pyloric rhythm (Marder and Selverston, 1992), which has been statistically inferred in other studies (Prinz et al., 2004; Gonçalves et al., 2019).