Main

A fundamental barrier to resolving a causal model of the nervous system is that causal relationships in the brain cannot be inferred solely from passive measurements of neural activity4,5. Direct perturbation of neural activity (for example, optogenetic stimulation) confronts this problem and has therefore been an area of intense methodological research and resulting progress. However, a clear approach for how to use these tools to obtain a causal model of neural activity has not emerged. Here we introduce a combined statistical and experimental strategy and demonstrate that it can efficiently learn a causal model of the fly brain.

We adapt a technique known in the statistical literature as ‘instrumental variables’6 (IVs). This technique was developed to estimate causal relationships in observational data in which direct experimental control is unfeasible. It relies critically on the stringent requirements of an IV: that it only directly affects an observed variable that putatively affects the outcome of interest—in the absence of that effect, the IV is independent of all variables. The observed relationship between the IV and the outcome is then strictly a result of the causal effect of interest. Yet, despite the stringency of these requirements, optogenetic stimulation plausibly meets them7,8. Optogenetic stimulation affects neural activity, is independent of neural activity because it is controlled by the experimenter, and acts on the brain only through neurons that express opsins. Thus, the IV approach could in principle be used to estimate causal effects between neurons.

However, there are two fundamental problems with the IV approach applied to the entire nervous system of the fly. First, naively estimating effects between every pair of neurons in the fly (approximately (105)2 pairs) would require intractable amounts of data. This problem is an even more insurmountable barrier to learning causal models of organisms with larger numbers of neurons (for example, (108)2 potential effects for mice9). Second, it would be unfeasible in the fly—and in most organisms—to independently stimulate and record from all neurons at once. The effectome would thus need to be gradually constrained across experiments on small sub-populations. It is unclear how to order experiments such that insights into whole-brain dynamics are achieved efficiently. Here we show that the Flywire connectome1,2,3 provides a feasible path to surmounting both barriers in the fly.

First, a principled approach to improving the data efficiency of an estimator is to place priors on the parameters being estimated. The fly connectome can be used as a prior on effects between neurons in the fly brain: neurons with no synaptic contacts are unlikely to directly affect each other. The connectome of the fly is exceedingly sparse (around 0.01% of neuron pairs form a synaptic contact10): thus, a strong prior can be placed on the vast majority of interactions in the effectome. Furthermore, with synaptic counts and synapse-level electron microscopy-based neurotransmitter predictions provided by the Flywire connectome11 (see Methods, ‘Construction of fly connectome matrix’), plausible priors can be placed on the sign and magnitude of effects between the neurons that are connected (the confidence of those predictions can naturally be incorporated into the strength of the prior).

Second, because the connectome is a complete connectivity map of the fly brain, it is uniquely suited to guide efficient estimation of the effectome. Neuroscientists tend to study a given neural circuit because it has been studied previously. A data-driven approach to proposing neural circuits of interest could reveal important computations that had not yet been considered. Ideally such a method would discover independent neural circuits and rank them by their total effect on the brain. To predict which neural interactions contribute most strongly to whole-brain dynamics, we analyse the eigenmode decomposition of the connectome. The eigenvectors of the connectome, ordered by the magnitude of their eigenvalues, provide sets of neurons, and patterns of neural activity therein, that under our ‘connectome prior’ are predicted to have the greatest total effect on the brain.

We thus provide an experimentally tractable paradigm for learning a complete causal model of the fly brain that is uniquely enabled by the properties of the whole-brain connectome. Specifically, we show that the sparse connectivity between neurons markedly improves the efficiency of estimating causal effects. We also show that that small populations of neurons underlie dominant dynamical modes, suggesting that whole-brain dynamics can be constrained with a series of highly targeted—and thus feasible—experiments.

Here we first lay out an experimental setup and associated statistical model of neural activity in the fly. Next, we outline how to use optogenetic perturbation as an IV to infer the direct causal effects between neurons in the context of a linear dynamical system. We then show how, in the presence of confounds, the classic regression estimator will return biased results, whereas the IV approach gives a consistent estimator. Using a simulation of whole-brain neural activity based on the connectome, we demonstrate that the IV approach provides consistent estimates of ground truth neural connectivity. We find that because there is a massive number of potential downstream neurons from any given neuron, the standard IV regression estimator converges very slowly. This motivates use of the connectome to formulate a prior on the IV weights so that the estimator remains consistent (that is, even if the prior is wrong, the estimator will converge to ground truth with enough data) but orders of magnitude more efficient—to the degree that the prior is correct. Finally, we analyse our ‘connectome prior’ to reveal thousands of proposed circuits ranked by their predicted total effect on the brain. We analyse two of these circuits and find that one recapitulates a proposed circuit for computing opponent motion and the other provides a dynamical mechanism for visual spatial selectivity.

Our motivating setting is an optogenetic experiment in the fly (Fig. 1a): the activity of a population of ‘target’ neurons is observed, a subset of these are ‘source’ neurons that express opsin driven by nl independent lasers, and the remaining neurons remain unobserved—these can also be unobserved non-neuronal processes recurrently interacting with the observed neurons.

Fig. 1: Schematic and simulation of effectome inference using IVs.
figure 1

a, Example of a single fly experiment. Some sets of neurons in the fly are observed (black circles within the field of view of the microscope), and a subset of these express opsin and are being stimulated (neurons under red light) with a white noise pattern (red trace). Another set of neurons may be unobserved. All neurons can have directed interactions via synapses (arrows), and only the effects of stimulated neurons can be estimated (black arrows connecting stimulated neurons to observed neurons). b, In this setup, the laser can be cast as an instrumental variable (IV) that directly affects only the opsin-expressing neurons (arrow labelled α from L to X). Stimulated neurons in turn have a direct effect on downstream observed neurons (arrow β from X to Y), but common unobserved inputs (Z) may corrupt attempts to estimate the direct relationship. The IV approach uses the joint relationship between the laser and downstream neurons to determine the direct effect of the stimulated neurons on postsynaptic neurons. c, Top, a simulated example of confounding effect Z, which is given by a slow drifting signal (smooth green trace). Raw, uncorrupted responses of neurons X and Y are independent (pink and brown traces, middle), meaning that there is no connection from X to Y. Bottom, observations X and Y after adding the confounding signal Z, resulting in substantial correlation. d, The least-squares estimate of the weight from X to Y (blue line; mean ± s.d.; n = 100 simulations) exhibits large bias regardless of sample size, whereas the IV estimate (orange line) converges to the true effectome weight of zero (black dashed line).

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The graphical model that we associate with this setup identifies the lasers as the IVs that independently drive the source population (X) via a linear transformation (α), the source population drives the target population (Y) via β (the effect), and both receive common inputs from unobserved confounders Z (Fig. 1b). The fundamental difficulty of fitting neural models to observed data is that even if β = 0, Z can induce spurious dependence between the source and target population.

IVs are robust to unobserved variables

The principal challenge to inferring causal effects solely through passive observation of neural activity is the possibility of unobserved confounding inputs. There are many potential unobserved confounders in the fly. Typically, a small subset of the nervous system is imaged at a time: thus, input from unobserved neurons could confound the observed neurons. Whole-brain imaging is possible in the fly12,13, but resolving single-neuron dynamics remains a challenge owing to the density of the neuropil. Even if single neuron activity could be resolved, afferent activity from the peripheral nervous system and ventral nerve cord could potentially be a source of common variability. Moreover, neuropeptide signalling cannot be inferred from calcium imaging data alone14,15. Finally, measurement error, including physiological artifacts such as brain movement16, can act as a source of common variability. Collectively, there is a high probability of unknown confounding variables preventing valid causal inference during passive observation of neural activity in the fly brain.

A simple case that demonstrates the effect of confounding variables is one in which two neurons X and Y have no causal effect on each other (β = 0 between X and Y; Fig. 1b), but there is a common unobserved input to both from Z (for example, an unobserved neuron). Even though there is no causal effect of X on Y (Fig. 1d, ground truth weight is 0), the least-squares estimate incorrectly converges on a positive weight, whereas the IV converges on 0 because there is no correlation between the laser and Y. Thus the IV, under our assumed model, is not corrupted by unknown, unobserved inputs.

IV accurately estimates fly effectome

To demonstrate that the IV can, in principle, estimate the fly effectome, we applied it to a simulation of the entire fly brain during stimulation of a single source neuron and a whole-brain recording (see Methods, ‘Simulations to evaluate estimators’; for a graphical model, see Extended Data Fig. 1). We set our ground truth causal effect matrix entries to be proportional to the number of synapses multiplied by the ‘sign’ of those synapses—whether they were inhibitory or excitatory (Fig. 2a). We found that the estimator on average was accurate (Fig. 2b, scatter centred around diagonal), but the bulk of error came from estimates of weights whose true value was zero. This was because the vast majority of weights are zero owing to the sparsity of the fly connectome. Non-zero estimates of these weights were thus the dominant contribution to total estimation error.

Fig. 2: Inferring the effectome using standard and Bayesian approaches on simulated data.
figure 2

a, We used the connectome to set the effectome weights for a whole-brain simulation using all 121,327 connected neurons (neurons with no incoming or outgoing connections, above a threshold of >5 synapses, were not included). b, Synaptic weights were set proportional to synapse count, with positive (negative) sign for excitatory (inhibitory) synapses. c, IV estimates of postsynaptic weights of a single example neuron. Most of the error falls along the vertical line where true weight equals 0 (which is the majority of the weights, owing to the sparsity of the fly connectome). d, Mean ± 2 s.d. of an independent Gaussian connectome prior on each weight in the effectome. We set the prior mean to be proportional to the signed synapse count of the connectome and variance equal to the absolute value of the mean plus a small constant, so that the prior width is non-zero between neurons with no known synapses. e, An IV–Bayes estimator shows smaller error than the raw IV estimator in b. f, The error of estimated weights (residual sum of squares, RSS) decreases with the number of samples for both estimators, but IV–Bayes gives several orders of magnitude faster convergence (orange below blue line; mean ± s.d.; n = 10 simulations). Mean squared error will decrease indefinitely for both estimators because they are consistent (that is, they converge to ground truth as the number of samples goes to infinity). Horizontal lines show error level where the R2 of the recovered weights is zero (long dash) and 0.9 (short dash), respectively.

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The connectome as an effectome prior

The high variability of IV reflects a fundamental problem in fitting a model of the entire brain: the number of parameters is large. Yet, because the fly connectome is available and—critically—it happens to be highly sparse, we can markedly increase the efficiency of our model estimate by assuming that neurons with no synaptic contacts are unlikely to directly affect each other. We do so in a principled manner with an extension of our estimator to a Bayesian setting.

We took a Bayesian approach to reduce error by placing a Gaussian prior on the model weights (IV–Bayes). The Gaussian prior mean was proportional to the synaptic count and sign (positive for excitatory synapses and negative for inhibitory synapses) (Fig. 2d). The variance was equal to the absolute value of the mean plus a small constant. Thus, weights between non-anatomically connected neurons are strongly biased towards zero, whereas weights for connected neurons are weakly biased toward the scaled synapse count, allowing them to be zero if warranted by the data. We add a small constant to the prior variance so that the estimator remains consistent: even if the prior is wrong (for example, if a synapse exists where none was found in the connectome), the estimator will converge to ground truth with enough data.

We evaluated the IV–Bayes estimator using a simulated dataset generated with effectome weights set to a corrupted version of the ground truth connectome weights, thus creating a mismatch between the effectome weights and the prior mean (Methods, ‘Simulations to evaluate estimators’). This corruption could, for example, reflect natural variation between flies’ connectomes. We note that the IV estimator is not meant to estimate the underlying connectome. Instead, the estimator approximates the linear effects between neurons—these effects may have a weak relationship with the connectome, and will probably depend on the state of the nervous system (see Discussion, ‘A broader definition of the effectome’). We found that the IV–Bayes estimates outperformed the standard IV estimator, even though the prior mean was corrupted (Fig. 2c,e). In particular, the high variability of the IV estimate for zero weights was quenched in the IV–Bayes estimate. Intuitively, if the connectome provides information about the strength of causal interactions between neurons, it should outperform standard IV.

To quantify the relative efficiency of the naive IV approach and IV–Bayes, we sampled effectome matrices as described above and then evaluated the average residual sum of squares (RSS) of the two estimates as a function of the number of samples (for example, duration of experiment). As expected, we found that the RSS of both estimators decreased with increasing samples (Fig. 2f, blue and orange trace slope downwards). Yet, we found that the RSS of the IV estimator is at least an order of magnitude higher than that of the IV–Bayes estimator across number of samples (blue above orange). In terms of fraction of variance explained, IV–Bayes explains the vast majority of variance for the maximal number of time samples (orange trace below dotted line on right) but the raw IV estimator is still too noisy to achieve a positive quantity of fraction variance explained (blue trace above dashed line) (Fig. 2f). Thus, in simulation, IV–Bayes provides at least an order of magnitude improvement in converging to the ground truth causal effects.

Our simulations thus far have focused on the stimulation of a single neuron while the entire fly brain is observed. If every neuron in the fly brain was stimulated independently while every neuron was being observed, the entire fly effectome would be identifiable within a single experiment (Extended Data Fig. 2b; for multi-neuron simulations see Extended Data Figs. 3–6). However, it is unclear whether this approach is experimentally feasible, given the constraints of diffraction-limited optics, which makes independent stimulation and identification of every neuron in a whole-brain imaging setup challenging. A more feasible approach would be to sparsely image and stimulate neurons to estimate the effectome across flies (Extended Data Fig. 2c). We now describe a data-driven strategy for systematically choosing subsets of neurons that account for disproportionate shares of neural dynamics.

The connectome reveals dominant circuits

Here we demonstrate a data-driven method for ranking sets of source neurons that are most likely to form circuits with a large effect on the fly nervous system. We propose that these circuits should be prioritized for interrogation by our estimator. Specifically, we consider a recurrent neural network model of whole-brain activity given by

$${{\bf{r}}}_{t+1}=W{{\bf{r}}}_{t},$$
(1)

where rt is a vector denoting the activity of all neurons at time t and W is the effectome weight matrix, which (for these analyses) we set to the scaled, signed synaptic counts extracted from the connectome. To analyse the fly brain’s dynamical properties, we perform an eigendecomposition of the weight matrix W, which decomposes global dynamics into patterns of neural activation called eigenvectors with simple two-dimensional rotational dynamics determined by an associated eigenvalue. Below, on the basis of this eigendecomposition, we provide testable hypotheses pertaining to both global properties of fly neuronal dynamics and highly specific circuits.

In the model we have proposed (equation (1)), the eigenvectors of the effectome describe the dominant modes or patterns of neural activity that will grow or decay over time, each governed by its associated eigenvalue. For example, if the activity of the brain at time 0 is set to the ith eigenvector vi (whose eigenvalue is λi), then the brain’s activity pattern at time step t is given by

$${{\bf{r}}}_{t}={\lambda }_{i}^{t}{{\bf{v}}}_{i}.$$
(2)

Thus, the magnitude of the eigenvalue precisely determines the magnitude and duration of the effect of this pattern of activity. The eigenvectors with the largest eigenvalues are therefore plausibly associated with neural dynamics that have the largest total effects on the fly brain. The neurons associated with the significant coefficients, or ‘loadings’, in an eigenvector indicate the sub-population of neurons whose connectivity principally sustains these dynamics, forming an ‘eigencircuit’.

There are two critical properties of the eigendecomposition that determine the rate at which neural dynamics can be constrained by the estimated effectome. The first is the sparsity of eigenvectors. If the pattern of activity specified by the eigenvector includes only a few neurons with non-zero loadings, then only the effectome of those neurons needs to be learned to specify that dynamical mode. By contrast, if each eigenvector significantly involves all neurons, the entire effectome would need to be learned to explain even one dynamical mode. The second critical property is the relative magnitude of the eigenvalues. If the eigenvalue associated with a sparse eigenvector was much larger than all the others, then the majority of variation in global dynamics could be explained with the effectome of a handful of neurons.

Global dynamics of putative effectome

We first examined the relative magnitude of the eigenvalues and found they decayed slowly. For example, the 1,000th eigenvalue has approximately 1/10 of the magnitude of the largest (Fig. 3a). This implies that: (1) the choice of which early modes to analyse is somewhat arbitrary, because they have similar magnitudes; and (2) many dynamical modes could be required to explain neural dynamics in the fly brain (that is, fly neural dynamics are high-dimensional). We note that the dimensionality of neural dynamics depends on the input distribution. This analysis implicitly assumes private white noise inputs to each neuron where all eigenvectors are equally likely to be driven—intuitively, this may correspond to a resting state with each neuron stochastically firing at a similar rate.

Fig. 3: Putative global dynamical properties of the fly central nervous system.
figure 3

a, Magnitude of the top 1,000 eigenvalues of the putative effectome (scaled matrix of signed synaptic counts extracted from the connectome) and power-law fit. b, Eigenvalues plotted in the complex plane. c, Correlation of eigenvectors sorted by associated eigenvalue magnitude. d, Median (black), 99th percentile (dark grey) and maximum (light grey) correlation between each eigenvector and all other top 1,000 eigenvectors, showing that the first 10–20 eigenvectors are nearly orthogonal to other eigenvectors and the correlation between other eigenvectors is highly sparse. e, Per neuron eigenvector loadings for the first (top) and 50th (bottom) eigenvector. f, Concentration of eigenvector loadings, quantified by the number of neurons needed to account for 75% of the power of the eigenvector. g, Anatomical renderings of neurons needed to account for 75% of the eigenvector’s loading power. Eigenvectors are drawn from the top 100 eigenvectors, and each column displays eigenvectors with neurons predominantly associated with either visual, olfactory or motor/navigation anatomical locations.

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We explored the range of timescales for predicted dynamics by examining the eigenvalues in the complex plane (Fig. 3b). Complex eigenvalues are associated with a complex eigenvector whose real and imaginary parts define a plane of rotational dynamics for neural activity. If a pattern of activity is induced within this plane, then neural activity will continue to evolve solely within the plane—that is, it will transition over time between mixtures of the real and imaginary part of the eigenvector. The angle from the positive real axis determines the speed of these rotational dynamics. At 0°, the eigenvalue is real positive and there are no oscillatory dynamics but a simple monotonic decay. A non-zero angle is exactly the angular step size of rotational dynamics at each time step, so small angles imply slow rotational dynamics, and a negative real eigenvalue (180°) implies the fastest possible rotational dynamics (flipping sign at each time step). We find there is a broad distribution of timescales of rotational dynamics. The distribution of angles had modes at 0°, 90° and 180°, implying a preponderance of monotonically decaying modes, rapid transitions between distinct populations and the fastest possible timescale, respectively.

We then investigated whether dynamical modes tended to be independent of each other. When eigenvectors are orthogonal to each other, the dynamics associated with each eigenvector are independent (under assumed white noise inputs). Conversely, if two eigenvectors are highly correlated, then these dynamics are more likely to co-occur. Thus, by examining the correlation between eigenvectors we can test which dynamical motifs will tend to be enlisted simultaneously—potentially because they are involved in similar computations. Examining the correlation matrix of the first 100 eigenvectors (treating the real and complex parts as separate eigenvectors), we found on average correlation was weak with sparsely distributed higher values (Fig. 3c). We observed that early eigenvectors tended to have lower correlation to others (first ten rows and columns dark). Aggregating the correlation of each of the top 1,000 eigenvectors to all others, we found a clear trend in which both the max and average correlation increased with eigenvalue rank (Fig. 3d). Roughly speaking, the top ten dynamical modes will occur independently of each other, whereas the rest will tend to co-occur with at least one other mode. Thus, we predict that dynamical modes will typically operate independently of each other, which bodes well for the project of examining these circuits individually.

Experimental limitations dictate that only sparse populations of neurons can be simultaneously identified with the connectome and independently stimulated. Thus, here we analysed whether the populations involved in these eigenmodes are in fact sparse. Plotting the first eigenvector, we found that the loadings across neurons were indeed extremely sparse, with most loadings near zero and only several deviating significantly from zero (Fig. 3e, top). These loadings were of the same sign, indicating that all neurons significantly involved in this mode oscillate in sync. We found that later modes were also sparse, but less so (Fig. 3e, bottom). We measured the number of neurons required to account for 75% of power across loadings for the first 1,000 modes, and found that this number never exceeded more than 10% of the fly brain (Fig. 3f). The dominant modes tended to be the sparsest, suggesting that the dominant dynamics of the fly brain can be explained by estimating only a small fraction of the effectome. On average, for the top 10 eigenvectors, around 50 neurons were needed (less than 0.05% of all neurons), for the 10th to 100th eigenvectors, around 500 neurons were needed (less than 0.5% of all neurons), and for the 100th to 1,000th eigenvector, around 1,500 neurons were needed (less than 1.25% of all neurons). These findings suggest that the dynamics of the dominant modes in the fly brain can be explained by estimating a small fraction of the effectome.

Finally, we characterized the anatomical properties of these putative circuits. We visualized up to 100 neurons that together comprise 75% of loading power within their respective eigenvector loadings and colour-coded the top 12 (the remaining neurons are in grey; by contrast, all results in Fig. 3a–e are for all neurons). To provide a broad sampling of circuits, we organized eigenvectors into three groups on the basis of anatomical location: visual, olfactory and motor/navigational (Fig. 3g, left, middle and right columns, respectively). In general, we found that the highly sparse top eigenvectors that we previously characterized were also anatomically localized. For example, the top visual eigenvectors contained neurons that were confined to the lobula plate in the left hemisphere (row 1) and right hemisphere (row 2) (Fig. 3g). These eigenvectors recapitulate a hypothesized neural circuit for opponent motion computation (Fig. 4a–f). The top olfactory eigenvectors were also anatomically localized and contained mushroom body neurons (row 1) and projection neurons from the antennal lobe to the lateral horn (row 2). Multiple motor/navigational eigenvectors also showed confinement to the ellipsoid body (rows 1 and 2). For all three anatomical categories, we observed that eigenvectors with lower sparsity tended to incorporate diverse cell types distributed across multiple neuropils (rows 3 and 4). In general, we found that neurons with high loadings in early eigenvectors were often anatomically localized in accordance with the classical approach of studying the nervous system region by region. Conversely, many circuits were not anatomically localized and merit further investigation (for examples, see Extended Data Fig. 7).

We observed that anatomically localized eigencircuits were in the minority (10% of the top 1,000 eigencircuits), and non-localized circuits were often among the most dominant (Extended Data Fig. 8e,f). We considered whether modest amounts of biological variability, measurement error or saturation would change these results (Extended Data Fig. 9), and found that in general the early eigencircuits were quite robust, and the later ones less so. The non-local circuits tended to be less robust to perturbation, but we did find many robust examples (Extended Data Fig. 9b). We also determined that the high dimensionality of the connectome was not simply a result of the sparsity of the connectome, nor was it sensitive to the aformentioned factors (Extended Data Fig. 10).

We found that the top eigenvectors of the connectome prior are highly sparse. This property facilitates learning the effectome because it is easier to identify, image and stimulate sparse populations. Given the thousands of existing genetic driver lines accessible to the fly community17 and tools for automatically screening these lines for neurons of interest18, it may be possible to identify reasonably sparse genetic lines that contain a subset of each eigencircuit’s dominant neurons (see Supplementary Information, ‘Sparse expression’). Altogether, owing to the distinct properties of the fly connectome—namely the sparsity of connections, eigenvector loadings and interaction between eigenmodes—there exists a plausible path forward to systematically and efficiently explain whole fly brain dynamics in terms of direct causal interactions between neurons.

Dominant circuits are interpretable

Our decomposition of the putative effectome revealed sparse eigenvector loadings, which makes them amenable to further analysis. We tested whether these vectors correspond to identifiable circuits and interpretable dynamics in the fly brain (see Methods, ‘Simulations to analyse eigencircuits’).

We found that the first eigenvector localized to the lobula plate (Fig. 4a) and was highly sparse (Fig. 4b). We quantified the anatomical localization as the fraction of synapses in a single neuropil. For this eigencircuit, 75% of synapses were localized to the lobula plate, whereas the rest were confined to the inferior posterior slope. The associated real eigenvalue was negative: thus, for linear dynamics we expect rapid oscillation, but in the more realistic case where activity is thresholded (see Methods, ‘Simulations to analyse eigencircuits’), these neurons will inactivate rapidly following activation (Fig. 4c). The top four neurons in this eigenvector were VCH, DCH, LPi15 and Am119. All putative effects between these neurons are inhibitory, but there is a complex mix of mutual and directed inhibition (Fig. 4d, partial symmetry across diagonal of weight matrix). Notably, VCH and DCH do not inhibit each other, but are inhibited by Am1 and LPi15, but Am1 receives recurrent inhibition from VCH and DCH, whereas LPi15 does not (Fig. 4e).

Fig. 4: Example eigencircuits.
figure 4

a, Neurons with top five loadings on eigenvector 1. b, First eigenvector of the fly effectome. c, Linear and rectified dynamics upon stimulation by first eigenvector (a.u., arbitrary units). d, Synaptic count and sign between neurons. e, Circuit diagram representation of synaptic weight and sign. f, Visual stimulation simulation. VCH and DCH are given sustained stimulation for 150 ms, simulating BTF motion on the right side of the fly (fly on the left; solid brown and yellow, high sustained response). All neurons were given sustained stimulation for 150 ms, simulating BTF motion on both sides of the fly, which leads to inhibited responses of VCH and DCH (fly on the right; dashed brown and yellow below solid). g, Neurons with top four loadings on eigenvector 45. h, All loadings for eigenvector 45. i, Linear and rectified dynamics (left) upon stimulation by eigenvector 45 (right). j, Synaptic count and sign between nine neurons with top loadings on eigenvector 45. k, Simulation of visual stimulation. Left, simulation of a 0° visual stimulus, with strong input to the 0°-preferring neuron (light blue trace) and sustained background input to all other neurons. The network response exhibits WTA dynamics, in which all neurons respond transiently to stimulus onset, but only the neuron with maximal response remains active for the entire stimulus duration. Right, similar results for a 90° stimulus and 90°-preferring neuron (green trace).

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Concretely determining the computation that this visual circuit may perform requires an assumption about how visual features drive these neurons. It is known that VCH and DCH receive major input from H2 from the contralateral eye, which responds to back-to-front (BTF) motion20, whereas LPi15 and Am1 receive major inputs from T4b and T5b, which are driven by ipsilateral BTF motion21. We probed the functional properties of this circuit by simulating BTF visual input (see Methods) to either the right eye (contralateral input) or the left and right eyes together (bilateral input). When provided with contralateral BTF input only, we observe that VCH and DCH activity remained high throughout the stimulation period, whereas LPi15 and Am1 activity were suppressed because they were not directly stimulated and because Am1 is strongly inhibited by VCH and DCH (Fig. 4f, solid trace brown and yellow above blue and green). Conversely, bilateral BTF stimulation resulted in suppression of VCH and DCH. Overall, this putative circuit is well-suited to compute opponent motion across the fly eyes.

Notably, this circuit was analysed in a very recent small-scale connectomic analysis of the optic lobe21, whereas it was ‘rediscovered’ with our data-driven approach. This suggests, anecdotally, that the eigendecomposition of the connectome can reveal scientifically interesting sub-circuits.

Inspired by our findings with the first eigenvector of the connectome prior, we next sought to identify additional circuits with a putative role in stimulus selection. We found that eigenvector 45 contained high loadings for GABAergic (γ-aminobutyric acid-producing) R4d ring neurons in the ellipsoid body (Fig. 4g). For this eigencircuit, 99% of synapses were localized to the ellipsoid body, and the remainder were in the fan-shaped body and mushroom body medial lobe. R4d ring neurons have spatial receptive fields that retinotopically tile the visual field and exhibit directional and orientation tuning22. This eigenvector has sparse and bimodal loading (Fig. 4h, majority of scatter near 0 but subset near 0.2). Its associated eigenvalue, similar to the first eigenvector, is negative real, leading to rapid oscillatory linear dynamics and inactivation for rectified dynamics (Fig. 4i). Despite the similar dynamics, the synaptic weight matrix revealed complete mutual inhibition between R4d ring neurons (Fig. 4j, off-diagonal blue). This connectivity pattern has been identified in prior work23, but its functional relevance remains incompletely understood.

One possible computation consistent with this connectivity pattern is a winner-take-all (WTA) computation with respect to visual features distributed across space. In a WTA circuit, the most strongly activated neuron strongly suppresses all other neurons, thereby preventing its own inhibition. To test this prediction, we simulated uniform visual drive to the R4d inhibitory sub-network while providing one neuron with higher input (Fig. 4k, left and Methods). We found that the neuron with a stronger input indeed had a robust sustained response, whereas the other neurons’ responses quickly decayed near to baseline (Fig. 4k). We confirmed that this WTA property is not specific to a single neuron by providing different neurons with biased input (Fig. 4k, right). Indeed, these dynamics persisted, which supports the idea that a WTA computation is a robust property of this circuit. On the basis of these findings, the R4d inhibitory sub-network appears well-poised to implement WTA dynamics and thus to select individual visual spatial channels as primary inputs to the central complex. We note that global mutual inhibition within ring neurons such as the R4d cell type has been characterized24 and predicted to potentially drive WTA, but here we demonstrate that a mechanism predicted directly from anatomical parameters generates this computation.

We have analysed two of the hundreds of sparse eigenvectors revealed by an eigendecomposition of the fly connectome. We emphasize that these eigenvectors should not be interpreted as the ‘true’ effectome eigenvectors; rather, they provide a principled approach for generating and testing falsifiable hypotheses about causal relationships between neurons and the computations they may support. Our analyses served to demonstrate how neurons may be systematically chosen for causal perturbation experiments, and how—once the true effectome weights are learned for this subset of neurons—one might generate hypotheses about neural function than can be tested in vivo. Crucially, the dynamical mechanisms of the computations indicated by our simulations remain untested and require estimating the effectome between these neurons.

Discussion

We developed a combined experimental and statistical approach to estimate a causal model of the fly central nervous system—its effectome. In simulation, we demonstrated that the approach provides consistent estimates of the ground truth effectome. We found that the huge number of parameters of a whole-brain model made this estimation unfeasible, motivating us to use the connectome as a prior to markedly increase the efficiency of our estimator. We analysed our connectome prior to reveal thousands of small putative circuits operating largely independently of each other. This indicates that whole-brain dynamics may be efficiently explained with sparse imaging and perturbation, which is far more feasible than dense imaging and perturbation. We analysed two of these circuits to find that one recapitulates a proposed circuit for computing spatial opponent motion, and the other provides an explicit mechanism for visual spatial selectivity in the ellipsoid body.

Related IVs work

IVs has been an area of intense interest outside of the neurosciences but was only recently recognized as a useful tool for neural perturbation analysis4,5,25,26. Non-parametric approaches that estimate average functional effects4,26 do not provide an explicit model of neural dynamics or differentiate between direct versus indirect synaptic effects. Yet, these non-parametric estimates are naturally more robust and could be used to validate model-based predictions. To our knowledge, our work is the first to provide a consistent estimator of a neural dynamics model in the challenging but nearly universal experimental conditions where there are potential unobserved confounders (for example, unobserved neurons).

An extension of this approach to higher order auto-regressive models would relax the restriction that the timescale of interaction is known and fixed (1 ms in our simulations); thus, slower effects (potentially through extra-synaptic paths such as peptide signalling pathways) could be detected. There has been recent progress in this direction27, but it does not currently allow recurrent interaction between observed and unobserved populations, which is typical in neuroscience settings.

Related connectomic work

There has been sustained interest in the analysis of large-scale anatomic information10,28,29,30. A linear systems analysis of the worm30 made several findings that are qualitatively similar to ours: they ‘rediscovered’ several known circuits, found a preponderance of fast oscillatory and monotonic decaying modes, and identified sparse eigencircuits (see fig. 8 in ref. 30).

Graph-based fly connectome analyses14,29 find that the fly brain is a small-world network with short paths between almost any pair of neurons, which could imply highly effective global communication. By contrast, we find that neural dynamics are best described by small, independent subsets of neurons: stimulating one eigencircuit has little effect on neurons outside of that circuit. Future perturbative work is required to determine the efficacy of global information propagation across the fly brain.

Along a similar line, a clustering algorithm has been applied to the fly hemibrain connectome28,29. It also recovered well-studied circuits, indicating its promise. Yet, the dynamics models under which the recovered circuits do in fact sustain distinct computations is unclear.

Anatomical information has been used to constrain mechanistic whole-brain models of the worm fit to neural data31, and a similar model could be applied to the fly. A critical distinction of our approach is that it provides consistent estimates under unobserved confounders. Even for whole-brain recordings (which are now possible in the worm), where one might assume there are no unobserved variables, it is impossible to directly test this assumption. Yet this prior work directly demonstrates that connectomic constraints can improve the efficiency of model estimation—as we show in simulation.

In an alternative connectome-based approach applied to the visual system of the fly32, parameters were not constrained to neural activity but were instead optimized to perform discrimination of visual motion. By sharing parameters across circuits that are thought to perform similar functions, the efficiency of estimation was greatly increased. In a similar approach, IV–Bayes could be extended to a hierarchical Bayesian model that borrows statistical power across circuits and cell types that are hypothesized to have similar functional properties.

We were able to recapitulate experiments on the visual system by directly stimulating model neurons in the central nervous system (Fig. 4), but this required prior knowledge of how specific sets of stimuli affect the central nervous system. Incorporating the effects of stimuli on early sensory neurons (for example, ommatidia) is a critical direction for utilizing effectome estimates to predict sensory computations for novel stimuli (for example, in refs. 32,33). Linking estimated internal dynamics models to behaviour—a central goal of systems neuroscience—will require integration with models of how descending motor neurons actuate the body34,35,36.

A broader definition of the effectome

The method that we introduce here is a consistent estimator of a linear dynamical system, yet the fly nervous system is a highly nonlinear system. To provide a concrete interpretation of linear model estimates applied to nonlinear neural dynamics, we analysed our estimator in the context of a conductance-based model of neural dynamics. This model includes both a spiking nonlinearity and nonlinear synaptic integration. We find that the IV estimate converges to the Jacobian (matrix of partial derivatives with respect to voltage of each neuron) of the underlying neural dynamics equation, evaluated at the voltages of the neurons in the population (Extended Data Fig. 3 and Supplementary Information, ‘Conductance-based neural dynamics model’). This result is entirely consistent with our notion of the effectome—the Jacobian captures the effect of a small perturbation of one neuron on any other neuron. Our linear simulation (equation (1)) is a special case in which the Jacobian is exactly the connectome. In the conductance model, the effect of a perturbation varies with the state of the network and the Jacobian captures this notion because it varies with respect to neuronal voltage. This also provides motivation to titrate optogenetic stimulation to the minimal amount possible—for larger deviations, the estimate of the Jacobian becomes less precise and averages over more states. The Jacobian is also a foundational quantity in the study of stochastic differential equations. It is classically utilized in a linear dynamical system that approximates nonlinear and non-stationary dynamics along an estimated trajectory. For example, in the case of neural data, independent IV estimates can be formed across a stimulus-triggered average. More generally, extensions to our estimator can utilize estimated trajectories within repeated experimental conditions, stereotyped behaviour or inferred latent states37. The latter may be critical if the emerging observation that the bulk of variation in global brain dynamics is unrelated to stimuli or behaviour holds in the fly38,39,40. Ultimately, we interpret effectome estimates as capturing local interactions between neurons in a particular state (for example, at a set of membrane voltages). Further research is needed to understand how to use these local estimates to learn about changes in state (such as synaptic plasticity).

Our interpretation of our estimator as providing the Jacobian of neural dynamics further clarifies the utility of the connectome prior. We show analytically in our conductance model that the Jacobian between unconnected neurons will always be zero (Supplementary Information, equation 6). Thus, the connectome prior mean will be wrong for only 0.01% of the effectome parameters, because only 0.01% of pairs of neurons form a synapse in the fly brain. Thus, even with misspecified non-zero weights IV–Bayes provides a large gain in statistical efficiency (Extended Data Fig. 4). In general, most neurons in the fly brain do not directly affect each other, and this is where the majority of our gains in statistical efficiency are achieved. This in combination with weak priors on the small subset of neurons that plausibly directly affect each other allows our estimator the flexibility to efficiently estimate the effectome. We also confirm that even with a completely misspecified prior (for example, there may be extra-synaptic effects through peptide signalling pathways41), our estimator, with enough data, converges to the Jacobian—that is, the weight matrix that provides a linear description of the effect of each neuron on any other in the system’s current state (Extended Data Fig. 5).

The connection of the effectome to the Jacobian of conductance model also provides a concrete interpretation of the connectome (matrix of signed synaptic counts) as a linear dynamics matrix. It is in only a narrow set of situations that the best linear approximation to these nonlinear neural dynamics would be proportional to the connectome. Variation in neural voltage across neurons, synaptic reversal potentials, synaptic conductance, membrane time constants and more can all corrupt proportionality (Supplementary Information, equation 6). Thus, a biophysical interpretation of our results is that the eigencircuits decompose the best linear approximation to neural dynamics under the assumption that there is a small amount of variation across these neuronal properties. A mild confirmation of this untested assumption is that we do in fact recover known functional sub-circuits of the fly nervous system from an eigendecomposition of the connectome (Fig. 4).

In general, the form of dynamics in the fly brain even for our simplified conductance model is highly under-constrained—the parameters needed to evaluate the Jacobian are not available (including average voltage, membrane time constants, postsynaptic currents associated with different synapse types and morphologies). It is beyond the scope of this work to identify biophysically realistic models of neural dynamics from the connectome and measured neural activity. Our estimate of the effectome represents a first-order approximation to these underlying dynamics, which will depend on the state of the nervous system. Future work could focus on how sets of these first-order estimates can be used to infer the appropriate models and parameters of nonlinear neural dynamics equations.

Sufficient experimental setting

Here we specify the sufficient, feasible, experimental setting to ‘learn’ the effectome. We mean feasible in that the technologies to perform these experiments exist but have yet to be simultaneously integrated into the fly. In Supplementary Information, ‘Experimental approach’, we discuss how to achieve this ideal setting, including practical steps for generating a fly line specific to estimating an eigencircuit.

We note that these requirements are sufficient but that there could certainly be alternative approaches to satisfying the requirements of our estimator. In general, we expect that it would be unfeasible to estimate a large fraction of the effectome in a single experiment. Thus, our proposed strategy is to estimate eigenvectors of the effectome piecemeal, with sparse subsets across genetically identical flies in the same experimental conditions (Supplementary Information, ‘Experimental approach’).

There are four main technical requirements to estimate the effectome in such a manner. (1) The ability to select a sparse subset of neurons within a population of interest to image and stimulate (Supplementary Information, ‘Sparse expression’). (2) The ability to record the intracellular voltage of a population of neurons (Supplementary Information, ‘Voltage imaging’). (3) Simultaneous independent direct stimulation of neurons in the population (Supplementary Information, ‘Holographic stimulation’). Both the recording and stimulation must be at the timescale of neuronal interaction. (4) Identification of imaged and stimulated cells with the connectome (Supplementary Information, ‘Identification pipeline’).

This ideal experimental setting is challenging to achieve. Nonetheless, if it is achieved, it provides a principled approach to accurately estimate a causal account of local neuronal dynamics. A complete nonlinear dynamical account of brain dynamics would necessarily recapitulate local dynamics; thus, the effectome will serve to rigorously constrain and test a mechanistic model of the nervous system.

Methods

Statistical model of fly brain

Here we approximate fly brain dynamics as a first-order vector auto-regressive model (VAR(1)),

$${{\bf{r}}}_{t}=W{{\bf{r}}}_{t-1}+{W}_{l,x}{L}_{t}+{{\epsilon }}_{t}.$$
(3)

where rt is the D × 1 vector of responses from all neurons on time step t, W is the full D × D dynamics matrix, Lt is the random Dl × 1 vector of stimulation power (for example, voltage), Wl,x is the matrix specifying the effect of stimulation, and ϵt is additive noise with arbitrary bounded covariance (for graphical model, see Extended Data Fig. 1). The raw IV estimator that we outline below requires no distributional assumptions on the random variables (Lt, ϵt). Furthermore, the noise (ϵt) can take on any correlation structure over time and across neurons, so our estimator applied to dynamics of this form will converge to the true effectome W, regardless of confounding inputs (so long as the noise is independent of the stimulation Lt). We note that for nonlinear dynamics, further assumptions are required (see Supplementary Information, ‘IV estimator applied to nonlinear dynamics’).

For our derivations below we define three subsets of neurons in rt, we will call the vector of neurons that are observed in a given experiment, target neurons, at time point t, Yt, the neurons that are both stimulated by the laser and observed Xt, source neurons, and finally neurons that are neither observed nor stimulated Zt (for graphical model, see Extended Data Fig. 1). Note that source neurons are a subset of target neurons.

Simulations

Simulations to analyse eigencircuits

To evaluate neural dynamics associated with eigencircuits (see ‘Dominant circuits are interpretable’) in continuous time, we restricted our analyses to neurons with the highest eigenvector loadings that accounted for 75% of all eigenvector power. In the case of the opponent motion circuit, this was 5 neurons, and in the case of the ellipsoid body circuit, this was 21 neurons. We set the discrete time step to T = 10 ms, and a sampling period of Δ = 1 ms so that at each time step,

$${{\bf{r}}}_{t+\varDelta }=\exp \left(\log \left(W\right)\varDelta {T}^{-1}\right)f\left({{\bf{r}}}_{t}+{{\bf{u}}}_{t}\right)$$
(4)

where W is the putative effectome of the subset of neurons being simulated, f(·) is a nonlinearity (here rectification), and ut is a vector of time-varying inputs.

We tuned the membrane time constant manually to 0.5 ms to recapitulate the qualitative findings of prior work. The resulting W was scaled to have a maximum eigenvalue of 1. It is unknown what a good approximate scaling of synapse number to linear effect is in these regions but if the scaling was zero then this circuit would not perform a computation and if it was too large the dynamics would be unstable. The membrane time constant is often faster in sensory systems but depends on the voltage of the neuron42. Future work could consider other normative motivations for scaling weights and time constants (for example, ref. 32).

In both the ellipsoid and opponent motion simulations, inputs in Fig. 4 were simulated by adding in a step function input to the relevant neurons in our dynamics simulation. This step function was either at 0 or a hand tuned maximum. In the opponent motion circuit input maximum was 0.01 and for the ellipsoid body circuit the winner neuron receiving a larger input had input of 0.01 and other neurons had 0.006. In both cases stimulation lasted 150 ms. In the case of the opponent motion circuit, we used prior literature to determine which neurons to stimulate under unilateral and bilateral BTF motion20,21. In the case of the ellipsoid body circuit, the ordering of the neurons with respect to retinotopic input was arbitrarily set by their eigenvector loading magnitude.

Simulations to evaluate estimators

In all simulations to evaluate statistical estimators, \({{\epsilon }}_{t} \sim {N}_{D}\left(0,{\varSigma }_{{\epsilon }}\right)\) where \({{\Sigma }}_{{\epsilon }}=c{I}_{D}\) and \({L}_{t} \sim {N}_{D}\left(0,l{I}_{D}\right)\). To simulate misspecification of the connectome prior mean, we estimated the accuracy of our estimator across many ‘ground truth’ effectomes drawn from the connectome prior (except without a small constant added to the variance so that synaptic weights equal to 0 remained 0), such that the connectome prior mean was never the same as the ‘ground truth’ effectome in a given simulation.

Given that there are a host of unknowns with respect to a real experimental setting (such as imaging SNR, strength of laser effect or duration of recordings), we hand tuned these parameters to give reasonable rates of convergence. In our whole-brain IV simulations, signal-to-noise ratio (SNR) = 10 (laser power relative to noise power). We note that while we have simulated from a parameter regime in which our estimator converges rapidly, there are many parameter settings where convergence is slow. We note that conditions of high noise and little effect of the laser are particularly challenging. Slow timescales are also challenging because they effectively filter out most of the power of the white noise perturbation (but for extension to correlated IVs, see ref. 27). Yet, we show analytically that our estimator is consistent and thus will converge with enough samples.

For clarity, in our example simulation we chose a single neuron to stimulate and estimate its downstream synaptic weights (Fig. 3). This neuron was chosen because it had a larger than typical number of downstream contacts. It is straightforward to estimate downstream weights for multiple neurons simultaneously (equation (7)), and we demonstrate this both for IV and IV–Bayes (Extended Data Figs. 4–6) and IV in a conductance-based model (Extended Data Figs. 3 and 4). We note that in our simulations, as the number of independent perturbations increases (nl), there is effectively more noise overall both through second-order effects and in estimating the effect on a downstream neuron that itself is being perturbed. This is another pressure to keep the number of stimulated neurons low and the strength of perturbation minimal, but depends on the particularity of connectivity.

IV estimator for an LDS

We note that

$$\text{Cov}\left[{X}_{t},{L}_{t}\right]={W}_{l,x}$$
(5)

because the stimulation is assumed to have identity covariance. Thus, by calculating the sample covariance between the laser and simultaneous activity in the stimulated neurons we can obtain an unbiased estimate of the linear weighting of laser drive on each neuron. Similarly, we can obtain an unbiased estimate of the linear effect of the laser on all target neurons at the next time step,

$$\text{Cov}\left[{Y}_{t+1},{L}_{t}\right]={W}_{x,y}{W}_{l,x},$$
(6)

where Wx,y is the submatrix of W with postsynaptic effects of Xt on Yt + 1. We can then use equation (5) to identify Wx,y with

$${W}_{x,y}={W}_{x,y}{W}_{l,x}{\left({W}_{l,x}\right)}^{+},$$
(7)

where (Wl,x)+ is a pseudo inverse because we have not specified the rank of L. Only if nL ≥ nS is this a true inverse and Wl,x is invertible. An equivalent but more intuitive approach is termed two-stage least-squares (2SLS), where in the first stage Lt is regressed on Xt to give \({\hat{X}}_{t}={\hat{W}}_{l,x}{L}_{t}\) and then \({\hat{X}}_{t}\) is regressed on \({Y}_{t+1}\) to give \({\hat{Y}}_{t+1}={\hat{W}}_{x,y}\,{\hat{X}}_{t}\). The IV estimator can also be extended to higher order AR processes by conditioning the estimator on multiple past time steps27.

We note that multi-synaptic effects can be derived from the estimated monosynaptic effects with powers of the effectome matrix. For example, if we have the effectome matrix W and input r is an input vector of all zeros except for neuron i, then the nth order synaptic effect is exactly Wnr (for example, n = 1 gives direct synaptic effects, n = 2 gives effects through up to two synapses, and so on).

The connectome prior

We use the 2SLS approach to motivate a consistent estimator from a Bayesian perspective. In short, we perform classical Bayesian regression for the second stage of regression using the connectome as a prior on the weights Wx,y. To be consistent with the most typical setting of Bayesian regression we first work out the case of multiple source neurons and a single target neuron (that is, learning a set of weights in the same row of W). We assume the conditional distribution of the output given the input is:

$${y}_{t}{\rm{| }}{{\bf{x}}}_{t} \sim {\mathscr{N}}({{\bf{w}}}^{{\rm{T}}}{{\bf{x}}}_{t},{\sigma }^{2}),$$
(8)

where \(\left({{\bf{x}}}_{t},{{\bf{y}}}_{t}\right)\) represents the input and output for sample \(t\in \{1,\ldots ,T\}\), and \({\sigma }^{2}\) is the variance of the observation noise in y.

Let us now suppose that μ provides the mean for a Gaussian prior over the linear weights w:

$${\bf{w}} \sim {\mathscr{N}}(\mu ,{\gamma }^{2}I).$$
(9)

Let μ = sc, where the hyperparameter s scales c, the connectome prior we set to be equal to the synaptic count and sign. Combining this prior with the likelihood defined above gives us the following posterior mean:

$$\begin{array}{l}{\hat{{\bf{w}}}}_{{\rm{M}}{\rm{A}}{\rm{P}}}={\text{arg}\text{max}}_{{\bf{w}}}P({\bf{w}}|X,Y,\theta )\\ \,\,\,=\,{(1/{\sigma }^{2}{X}^{{\rm{T}}}X+1/{\gamma }^{2}I)}^{-1}(1/{\sigma }^{2}{X}^{{\rm{T}}}Y+1/{\gamma }^{2}\mu )\\ \,\,\,=\,{({X}^{{\rm{T}}}X+{\sigma }^{2}/{\gamma }^{2}I)}^{-1}{X}^{{\rm{T}}}Y+{({\gamma }^{2}/{\sigma }^{2}{X}^{{\rm{T}}}X+I)}^{-1}\mu ,\end{array}$$
(10)

where \(\theta =\{{\sigma }^{2},{\gamma }^{2},c\}\) denotes the hyperparameters. The second expression above (equation (10)) shows that the maximum a posteriori (MAP) estimate is the standard ‘ridge regression’ estimate, \({\hat{{\bf{w}}}}_{{\rm{r}}{\rm{i}}{\rm{d}}{\rm{g}}{\rm{e}}}={\left({X}^{{\rm{T}}}X+\frac{{\sigma }^{2}}{{\gamma }^{2}}I\right)}^{-1}{X}^{{\rm{T}}}Y\), plus a term that biases the estimate towards the anatomical connectome μ. Note that in the limit of small observation noise \({\sigma }^{2}\) or large prior variance \({\gamma }^{2}\), the MAP estimate converges to the maximum likelihood (ML) estimate, whereas in the limit of large \({\sigma }^{2}\) or small \({\gamma }^{2}\), it converges to μ.

In our simulations we choose the optimal hyperparameters beforehand but the hyperparameters could be learned via a standard cross-validation grid search. A more principled approach would be to use evidence optimization,

$$\hat{\theta }=\text{arg}\,{\text{max}}_{\theta }P(Y{\rm{| }}X,\theta )=\text{arg}\,\mathop{\text{max}}\limits_{\theta }\int P(Y{\rm{| }}X,{\bf{w}},\theta )P({\bf{w}}{\rm{| }}\theta )d{\bf{w}},$$
(11)

which would be straightforward given that the evidence is available in closed form for this model.

Construction of fly connectome matrix

The connectome is a reconstruction of the central nervous system of a seven-day-old adult female Drosophilia melanogaster. We use the most recent version of the connectome v783. Details of the reconstruction are provided in the original publications of the connectome dataset1.

Each entry in the connectome matrix W, the main object of study in our analyses, was the number of synapses multiplied by their inferred sign based on predicted neurotransmitter type11. Specifically, neurons with neurotransmitters acetylcholine and dopamine had positive weights on their downstream neurons and neurons with GABA, serotonin, glutamate and octopamine had negative weights. The neurotransmitter type was predicted directly from electron microscopy images trained on synapses with known neurotransmitter types. The matrix W scaled for stability was used as the connectome prior mean in estimator simulations (Fig. 2) and our eigendecomposition analysis (equation (2); Figs. 3 and 4). A threshold was set on the synapse count such that any connections with less than five synapses were set to zero. This choice followed the reasoning of other analyses of the connectome10 that this would minimize the impact of spurious synapses—manual proofreading did not extend to connections with fewer than five synapses.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.