Bayesian Mapping of the Striatal Microcircuit Reveals Robust Asymmetries in the Probabilities and Distances of Connections

Patch-clamp data on connection rates between SPNs

We begin by reviewing key data on the connections within and between the D1 and D2 type SPNs, which we will also use to motivate our Bayesian approach. Previous studies by Taverna et al. (2008) and Planert et al. (2010) collected data on pairwise connections between SPN subtypes in slices obtained from both the dorsal and ventral striatum of mice. The subtype of the SPNs was determined by targeted expression of EGFP under the control of either a D1 or D2 receptor promoter sequence for Taverna et al. (2008) and of only a D1 receptor promoter for Planert et al. (2010). Nonlabeled SPNs were then assumed to belong to whichever group was not meant to be labeled in this particular animal, and electrophysiological criteria were used to exclude interneurons. After choosing a pair of neighboring cells, no further apart than 50 μm, Taverna et al. (2008) detected connections in current-clamp mode by injecting a depolarizing current step in the first (potentially presynaptic) neuron of the pair, and measuring depolarizing postsynaptic potentials in the second one. This procedure was then repeated the other way around. Planert et al. (2010) recorded up to four neighboring neurons simultaneously, with a larger maximum intersomatic distance of 100 μm, and detected connections by stimulating one neuron with a train of depolarizing pulses to generate action potentials in the presynaptic neuron and recording the potential postsynaptic responses of the other neurons.

In both studies, the ratio of successful tests to the total number of tests (Table 1) was then reported as an estimate p̂ for the true probability of connection between these types of neuron (plotted as the bars in Fig. 1A,B), in accordance with frequentist inference about a proportion. As they are estimates, they come with a level of uncertainty about the true proportion that depends on sample size, which here is the number of pairs that were tested. In a frequentist approach, this uncertainty would usually be given by a confidence interval.

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Figure 1.

Probability of lateral connections between SPNs estimated using either frequentist or Bayesian methods. A, B, Frequentist estimates of the probabilities of connection computed from intracellular recording data, and our computed 95% Wilson confidence intervals. C, D, Posterior probability density functions for the probability of connection using a Bayesian approach. Colored bars underneath the plot represent the 95% credibility intervals corresponding to each probability density function. Inset, Shape of the prior, a uniform distribution. E, F, Posterior probability density functions using the Jeffreys prior. G, H, Posterior probability density functions using a prior based on previous literature with mean equal to 0.12 and variance equal to 0.005.

However, typically, intracellular recording studies do not report any estimate for the uncertainty surrounding their measurements of p̂, and recent theoretical studies of striatum (Burke et al., 2017; Hjorth et al., 2020; Bahuguna et al., 2015) have simply used the raw values p̂ from Taverna et al. (2008) and Planert et al. (2010) to construct their models, supposing in particular that the probability of D1 to D1 connections is about twice as large as D1 to D2 connections. When we do compute confidence intervals, such as the Wilson confidence interval for binomial proportions (Brown et al., 2001) that we add ourselves in Figure 1A, B, we find that, given the relatively small sample sizes, the confidence intervals overlap considerably.

Bayesian inference of connection probabilities

As we have just explained, frequentist inference gives us a single point estimate p̂ for the probability of connection, normally surrounded by a confidence interval, which may be too large to be of any practical use and also, because it is flat, may give the illusion that the true value of p might be anywhere within this interval with equal probability. By contrast, Bayesian inference is more informative because it gives us a full probability density function f_p(p), called the posterior, telling us exactly how likely every possible value of p actually is, given the collected data. In this way, even when confidence intervals overlap, as is the case for practically all the SPN to SPN connections here (Fig. 1A,B), which in a frequentist interpretation would lead us to dismiss the difference as nonsignificant without insight as to whether this is because of insufficient data or a true nondifference (Dienes, 2014; Makin and De Xivry, 2019), Bayesian inference gives us a much clearer picture of what the data can tell us. We introduce here a simple Bayesian approach to calculating the full posterior f_p(p) for each type of connection from any pairwise intracellular recording data.

As we show in Materials and Methods, for these data, Bayesian inference turns out to be simple. Given the number of pairwise tests n, and the number of successful connections k, the posterior for p is a beta distribution with updated shape parameters as follows: fposterior(p)=Beta(p;a + k,b + n−k),(34) given initial values for its two parameters a and b. These initial values define the prior distribution for p, which reflects our initial beliefs about the possible values of p.

For instance, if we had initial values of a and b equal to 1, and we were looking at the data concerning D1 → D1 connections obtained by Taverna et al. (2008) (Table 1) with n = 38 tests and k = 5 connections found, then we would obtain a = 6 and b = 34, and the resulting posterior would be the one depicted in Figure 1C (light blue curve). Depending on our assumptions, different values of a and b can be used to give the prior a desired shape. We begin with the common choice of the uniform distribution in which p could be anywhere between 0 and 1 with equal probability, achieved by setting a = b = 1 as in the example just given.

Using this prior in combination with the data of Taverna et al. (2008) gives us the posterior curves shown in Figure 1C. Once obtained, we can revert, if necessary, to a more frequentist standpoint by extracting from these density functions a single point estimate p̂, typically the maximum a posteriori (MAP) value, which is simply the value of p for which f_p(p) is maximum, and a credibility interval around that MAP, which is the Bayesian equivalent of a confidence interval. The MAPs concentrate around relatively low values of p; and their exact values, which are given in Table 2, lie between 0.06 for D1 → D2 pairs and 0.28 for D2 → D1 pairs. The uncertainty surrounding p is given by the width of the posteriors and of the 95% credibility intervals underneath the curves. For the data from Taverna et al. (2008), the connections with the smallest credibility interval of ∼0.1 are the D1 → D2 pairs, whereas the least well-resolved connections are the D2 → D1 pairs for which the credibility interval spans ∼0.2. By contrast, when we apply the uniform prior to the data of Planert et al. (2010), the D2 → D1 connections have the smallest uncertainty and the D2 → D2 connections the largest (Fig. 1D).

The fact that we used the same prior for all pairs of neuron types reflects our initial belief that there is no difference in the probability of connection between pairs. To overcome this belief requires a sufficient amount of evidence, and we can start comparing the different probabilities of connection visually by looking at how much the different posteriors overlap. Based on Figure 1C, for example, it seems that, in the data of Taverna et al. (2008), probabilities of connection segregate depending on the nature of the presynaptic neuron in the pair. The posteriors involving presynaptic D1 neurons overlap considerably with one another, and their region of highest density is lower than for presynaptic D2 neurons who also show great overlap, while there is much less overlap between pairs with different presynaptic neurons. This becomes even more obvious when looking at the 95% credibility intervals drawn underneath the curves that show more or less overlap depending on the nature of the presynaptic neuron: the credibility intervals for pairs with a presynaptic D1 neuron share an overlapping interval roughly covering probabilities of 0.05-0.15, while the overlapping interval for connections with a presynaptic D2 SPN ranges between probabilities of ∼0.20-0.28. A similar pattern repeats itself in the data of Planert et al. (2010) (Fig. 1D), although the exact values of the overlapping regions are shifted toward 0 compared with Taverna et al. (2008), an effect that is potentially because of the maximum distance of sampling as explained later. This opens the possibility that there is indeed an asymmetry in terms of probability of connection that is dependent on the subtype of the presynaptic neuron, with no or little effect of the postsynaptic target subtype, something we will explore more thoroughly later.

One of the main advantages of Bayesian inference is that it forces researchers to be explicit about their priors and gives them the opportunity to choose appropriate ones. In order to illustrate this, we applied three further priors to the experimental data. First, the so-called noninformative Jeffreys prior sets a = b = 1/2. An intuitive way of understanding this prior is to picture ourselves at the very beginning of the experiment, waiting for the result of the very first paired stimulation and recording. This test will either be successful or not, meaning that the shape of the prior should give most and equal weight to these two outcomes (Fig. 1E, inset). Figure 1E, F shows the posteriors that result from using this prior, and we can see how they are practically identical to the posteriors obtained with a uniform prior. This was also the case when using the Haldane prior for which a and b equal 0 (not shown).

Our third prior is based on prior data, for Bayesian inference also provides us with a principled way of integrating previous knowledge into the prior. Earlier work (Taverna et al., 2004; Czubayko and Plenz, 2002; Koos et al., 2004) quantified the rate of lateral connections between SPNs without distinguishing SPN subtypes and concluded that lateral connections occurred at a rate of ∼0.12. Using this information, we can design a beta distribution with a mean of 0.12 and an arbitrary variance of 0.005 (see Materials and Methods), which serves as our third and final prior shown in the inset of Figure 1G. Although the posteriors are more clearly different from the ones obtained with the uniform and Jeffreys prior, they still look very similar. Indeed, we find that the MAP values for a given type of connection (Table 2) are very close whatever the choice of prior, and the previous observation that the curves seem to segregate according to the subtype of the presynaptic neuron is valid in every case. On the other hand, because this prior is more informative than the two previous ones, uncertainty is reduced, as witnessed by the smaller credibility intervals. Given this robustness to the different priors, we shall henceforth exclusively use the prior based on previous literature when analyzing connections between SPNs.

D1 neurons make fewer connections than D2 neurons

We previously observed that D1 neurons seem to make fewer connections than D2 neurons without necessarily targeting one subtype over the other, based on how the posterior distributions appear to segregate by presynaptic subtype in Figure 1. We can go beyond this qualitative analysis by calculating a density function f_Δ for the difference between two probabilities of connection. For instance, if we are interested in the difference in the probability of connection between D1 → D1 and D1 → D2 pairs (Fig. 2A), using the posterior distributions f_D1→D1(p) and f_D1→D2(p), we can find the density function for Δ_{(D1→D1)– (D1→D2)} by the following: fΔ(D1→D1)−(D1→D2)(Δ=k)=∫max(0,−k)min(1,1−k)fD1→D1(p=x+k)fD1→D2(p=x)dx(35) with the bounds of the integral such that x + k lies between 0 and 1. We can then calculate the probability that Δ_{(D1→D1)– (D1→D2)} is smaller than 0 by integrating this distribution between −1 and 0 (or calculate if it is greater than 0 by integrating the distribution between 0 and 1). By contrast, the frequentist strategy would be to compute a p value giving the probability of getting an experimental result at least as extreme as the one observed assuming the null hypothesis of no difference in connection probabilities (i.e., Δ = 0), whereas the Bayesian approach allows us to calculate the probability that Δ is less than (or greater than) 0 given experimental results. Thus, whereas the p value tells us how surprising the actual data are if we accept the null hypothesis, the Bayesian approach can quantify precisely how unlikely the null hypothesis actually is.

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Figure 2.

Comparison of the probabilities of connections between different SPN combinations. A, Density function for the difference in the probabilities of connection in pairs with a presynaptic D1 neuron using data from Taverna et al. (2008). B, Density function for the difference in the probabilities of connection in pairs with a presynaptic D2 neuron using data from Taverna et al. (2008). C, Posterior density functions for the probabilities of connection collapsed according to the presynaptic neuron subtype. Bars underneath the curves correspond to the 95% credibility intervals. D, Density function for the difference in connection probability between pairs with a presynaptic D1 neuron and pairs with a presynaptic D2 neuron. E-H, Same as in A-D, using data from Planert et al. (2010).

We applied this method to answer the question: do SPNs of either kind preferentially target SPNs of a certain subtype? In particular, based on their point estimates of connection probabilities, Taverna et al. (2008) have claimed that D1 neurons prefer to connect to other D1 neurons than to D2 neurons, a claim used in the computational study by Burke et al. (2017). As evidenced in Figure 2A, E, which plots the density functions for Δ_{(D1→D1)– (D1→D2)} according to the data by Taverna et al. (2008) and Planert et al. (2010), respectively, this is not the case as both density functions include 0 among their most likely values. Indeed, the probability that Δ_{(D1→D1)– (D1→D2)} is smaller than 0 is equal to 0.19 and 0.30 in Taverna et al. (2008) and Planert et al. (2010), respectively. Similarly, we do not find any difference when looking at whether D2 neurons have a preference for a particular postsynaptic neuron subtype (for the density functions of Δ_{(D2→D2)– (D2→D1)}, see Fig. 2B,F): for Taverna et al. (2008), the probability that Δ_{(D2→D2)– (D2→D1)} is larger than 0 is 0.16, and the probability that it is smaller than 0 is 0.17 for Planert et al. (2010). We thus find no evidence that SPNs of one subtype (D1 or D2) preferentially target a certain subtype.

Having established that rates of connection are not different for a given presynaptic neuron subtype, we can collapse the data according to the subtype of the presynaptic neuron to answer another question: are D1 neurons more or less likely to make connections overall than D2 neurons? To do this, we simply add up the total number of tested pairs and connections found for the same presynaptic neuron type (e.g., for D1 SPNs: nD1→SPN=nD1→D1+nD1→D2 and similarly for k). In essence, this is equivalent to considering the posterior of one connection rate as the prior for connections with that same presynaptic neuron subtype (e.g., f_D1→D1(p) is the prior for f_D1→SPN(p)). Figure 2C, G shows the posterior distributions for the collapsed datasets, and both studies agree that D1 SPNs are less likely than D2 SPNs to make connections to other SPNs. The MAP values for connection rates from D1 neurons are 0.092 and 0.059 in Taverna et al. (2008) and Planert et al. (2010), respectively, versus 0.199 and 0.143 for connection rates from D2 neurons. If we look at the density function for the difference between the probability of connections for D1 and D2 neurons (Fig. 2D,H), the MAPs for fΔ(D1→SPN)−(D2→SPN) are −0.11 and −0.08 for Taverna et al. (2008) and Planert et al. (2010), respectively, while in both cases, the probability that Δ_{(D1→SPN)– (D2→SPN)} is less than 0 is equal to 0.99. Thus, both studies contain very convincing evidence that D2 neurons are about twice as likely as D1 neurons to make connections to another SPN.

Probability of connection as a function of distance

So far, when considering data on SPN connections from Taverna et al. (2008) and Planert et al. (2010), we have been careful to analyze each study separately, resisting the temptation of combining the two into a more powerful dataset. We were justified in being so careful since the two experiments used different maximum intersomatic distances between neurons, namely, 50 μm in the study of Taverna et al. (2008) and 100 μm in that of Planert et al. (2010). Given that probability of connection between neurons typically decreases with distance (Hellwig, 2000; Humphries et al., 2010), sampling within a larger area around a neuron will probably cause a decrease in the ratio of connected pairs. Indeed, if we compare the probability of D1 or D2 neurons making lateral connections between the two experiments, we find that these probabilities tend to be larger in Taverna et al. (2008), which has the smaller sampling area, as illustrated by the corresponding density functions shown in Figure 3A. We thus introduce here a method for estimating the distance-dependent probability of connection between neuron types from data on neuron pairs recorded between some known maximum separation; our first use of this method is then to see whether the distance dependence is consistent between the two studies of SPN connectivity.

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Figure 3.

Estimating the probability of connection as a function of distance. A, Density function for the difference in connection rates for a given presynaptic SPN type between the Taverna et al. (2008) and Planert et al. (2010) studies. B, Experimenters chose their neurons within a certain maximum distance R_max, which defined a thin cylindrical volume of interest (here we draw the top of that cylinder). In the case of equiprobable sampling, the probability of choosing neurons further away increases as the infinitesimal volume corresponding to that distance increases as a linear function of r. C, The probability of finding a connected pair of neurons depends on two different processes: (1) the process of connection, modeled by the probability of connection between two neurons given the distance between them, which we postulate decays exponentially; and (2) the process of sampling neurons in the experiment, modeled as the probability of selecting another neuron at a given distance from a starting neuron. We explore here two different scenarios for the sampling process: an equiprobable scenario in which neurons within a determined volume are selected randomly, and a nearest-neighbor scenario in which the selected neuron is whichever is the closest within the maximum distance set by the experimenters. The overall rate of connection reported by the experimenters then corresponds to the integral (shaded areas) of the product of these two probability models. Hence, differences in sampling processes can cause different rates of connection, even if the probability of connection given distance is the same.

To do this, we start by positing that this decrease obeys a simple exponential decay function as follows: P(connection|distance=r)=e−βr(36) with β the decay parameter of unknown value, and r (for radius) the distance separating the two neurons. Ideally, to estimate this β parameter would require knowledge about the exact distance between every recorded pair of neurons, from which we could directly fit the model, but with simple assumptions on the sampling method used by experimenters, we can find an alternative way of converting values of β into p. Since the distance between each sampled pair of neurons in an experiment is indeed unknown to us, we shall consider it as a random variable. We can now express p as a function of β as follows: p(β)=∫0Rfsamp(r)e−βrdr,(37) which is the product of the probability f_samp(r) of experimenters selecting a neuron at distance r from another, and of the probability of these neurons being connected knowing r (Eq. 36) integrated over all possible values of r (for a visual depiction of what Eq. 37 means, see Fig. 3C). R is the maximum distance at which the experimenters are sampling neurons, equal to 50 or 100 μm in Taverna et al. (2008) and Planert et al. (2010), respectively. We now need to find f_samp(r).

A simple model for f_samp would be that, given a certain volume surrounding a central neuron, the probability of sampling any given neuron in that volume is equiprobable for all neurons (Fig. 3B): we call this model f_equi. With this assumption, we obtain the following solution (see Materials and Methods): p(β)=2R2∫0Rre−βrdr(38) which gives us the corresponding value of p for any desired value of β. As we have posteriors f_p(p) for the probability of connection between two types of neuron, we can now transform these into posteriors for β, f_β(β) through parameter substitution using Equation 38.

Probability of connection decreases faster for D1 than for D2 neurons

We apply this method to the posteriors for the probabilities of connection collapsed according to the subtype of the presynaptic SPN and obtain the posteriors for the decay rate β shown in Figure 4A, B for D1 and D2 neurons, respectively. Despite not being perfect, there is a good level of agreement between the two studies, which give estimates in the same ballpark. In both cases, although the posteriors do not overlap much, they do indeed lie quite close to each other providing us with a continuous, albeit broad, range of possible values. In the case of D1 neurons, the decay rate is expected to be in a region between 0.03 and 0.13 μm^–1. The exponential decay curves representing the probability of connection as a function of distance for a decay rate equal to the MAP of each study (which are given in Table 3) are also shown in the inset of Figure 4A, and it is evident that they are extremely close to one another. As for D2 neurons (Fig. 4B), the decay rate is smaller, as expected given that we have already shown that the overall probability of connection is higher for these neurons, ranging between 0.02 and 0.07 μm^–1.

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Figure 4.

Estimates for the distance dependence of connection probability between SPNs A, B, Posterior density functions for the decay parameter of an exponential function representing the probability that a D1 or D2 neuron connects to a neighboring neuron. Bars underneath represent 95% credibility intervals. Vertical black dashed line indicates the value of β at the maximum intersection of the two posteriors. Inset, Probabilities of connection given distance using the MAP values from the decay rate posteriors. C, D, Monte Carlo simulations in which the best intersection estimate of β from A and B is used to try and replicate the exact experimental results of Taverna et al. (2008) (left) and Planert et al. (2010) (right) concerning pairs with a D1 presynaptic neuron (C) or a presynaptic D2 neuron (D). The exact results obtained by the experimenters correspond to the red bars and given between brackets underneath the bar graphs. E, F, Density functions for the difference in decay rates between the two studies.

To get a better idea of how consistent the results are between the Taverna et al. (2008) and Planert et al. (2010) datasets, we used Monte Carlo simulations to try and recover the number of observations made by each experiment with its own value of R using the β value that seems most likely given the two posterior curves, that is, the intersection of the two posterior curves (Fig. 4A,B, black dotted lines). We ran 10,000 virtual experiments by generating random distances between pairs of neurons (according to Eq. 14) and the maximum distance R used by that study, and then deciding whether they were indeed connected according to the probability of connection of Equation 36. We generated the same number of pairs as tested in each study and then reported the number of times we obtained the exact same number of positive results (Fig. 4C,D, red bars in the histograms). For instance, setting an intermediate decay rate of 0.075 μm^–1 for D1 neurons, and generating 10,000 replications of the experiment of Taverna et al. (2008), which recorded 85 pairs of SPNs with a D1 presynaptic neuron, we obtained >500 simulations where exactly 8 pairs were connected, which is the number originally reported (Fig. 4C, left). In all 4 cases, we managed to replicate the original results relatively often, proving that the best estimates for β are reasonable. As a sanity check, we also used the decay rate MAPs for one subtype to try and replicate the results of the other subtype and found it much harder, if not impossible, to replicate the results, verifying that the decay rates have to be different for the two types of SPNs (results not shown).

If we compare the estimates between the two datasets more critically, there is a clear bias for the posterior curves extracted from Taverna et al. (2008) to be shifted to the right compared with the posterior curves from Planert et al. (2010) (Fig. 4A,B). Indeed, if we refer to the insets in Figure 4A, B, the best estimate of β according to the study by Taverna et al. (2008) would predict a 50% drop in probability of connection every 8 μm for D1 neurons versus every 13 μm according to the study by Planert et al. (2010). In the case of D2 neurons, the difference between the two exponential curves is even greater, with a half distance of 13 μm versus 21 μm according to Taverna et al. (2008) and Planert et al. (2010), respectively. Furthermore, the density functions for the difference in posteriors between the two studies both lie predominantly in the positive domain (Fig. 4E,F), and the probability that the decay rate is larger in the Taverna than Planert data is 0.967 and 0.996 for β_D1→SPN and β_D2→SPN, respectively. Although the disagreement between the datasets is small, it is nonetheless consistent.

Biased neuron sampling could explain differences between datasets

One potential explanation is that the sampling of neuron pairs was more complex than the equiprobable sampling model we first assumed. In this section, we explore this possibility by considering a model for f_samp where, rather than choose neurons equiprobably within a visible area surrounding a neuron, experimenters preferentially tested neurons that were closest to one another, to maximize the probability of detecting connections.

To explore this model, we used the same probability of connection given distance (Eq. 36) in combination with a new density function f_NN for the probability of the distance to the nearest neighbor, to derive a new mapping from p to β. We found the resulting mapping to be (see Materials and Methods): p(β)=k∫0Rre−r(πrhN+β)dr,(39) where k is a normalizing constant. Contrary to the previous equiprobable sampling model (Eq. 38), where these parameters cancelled out, this mapping depends on N, the density of SPNs in the striatum, and h, the height of the cylinder in which sampling takes place. We used here an estimate of the SPN density in mice of 80,500 per mm³, and tested three different values of h to get three different nearest-neighbor distributions: 0.1, 1, and 10 μm. We then use this mapping to transform the posteriors for p into posteriors for β as before (see Materials and Methods).

The first column of Figure 5 shows the resulting posteriors for D1 neurons. The picture for h = 0.1 μm is not so different from that obtained under the equiprobable sampling hypothesis, but for greater values of h, the posteriors overlap far more. In particular, for h = 1 μm, the posterior curves practically coincide. Similarly, for D2 neurons, h = 0.1 μm does not much improve the agreement between the two studies, but greater values of h do (Fig. 5, second column). This approach successfully illustrates how a tendency to select neurons closer together might account for the discrepancy observed in estimates of the decay parameter using the simpler equiprobable sampling model. Moreover, this analysis shows how the details of data sampling matter when estimating connectivity statistics from intracellular recording data.

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Figure 5.

Density functions for the decay parameter assuming a nearest-neighbor model of neuron selection for different values of the depth h of the sampling region. A, Density function for β for D1 neurons when h = 0.1 μm. B, Density function for β for D2 neurons when h = 0.1 μm. C, D, Same as in A, B for h = 1 μm. E, F, Same as in A, B for h = 10 μm.

Fast spiking (FS) interneurons preferentially connect to D1 SPNs

We now turn to completing our Bayesian map of the striatal microcircuit by evaluating the connections of different species of interneurons to the SPNs and to each other; we present the full map in Discussion (see Fig. 10). Three main types of interneurons are commonly documented (Kreitzer, 2009): FS, persistent low threshold spiking (PLTS), and cholinergic (ACh) interneurons. We will also include in this list TH and NPY-expressing NeuroGliaForm (NGF) interneurons because of their relationship to ACh interneurons, which is crucial to the function of the cholinergic component of the circuitry (English et al., 2011; Ibáñez-Sandoval et al., 2011; Dorst et al., 2020). We took data on pairwise intracellular recordings of these interneuron types from the range of studies listed in Table 1, and determined Bayesian posteriors for p as previously. Unlike for the SPNs, we did not have prior studies of the interneuron connections to help us design a prior, so we relied on a uniform prior instead.

We focus first on FS interneurons that project to the SPNs, using intracellular recording data from Planert et al. (2010) and Gittis et al. (2010). The posteriors we obtain for Planert et al. (2010) (Fig. 6B) are consistent with quite high connection probabilities: the MAPs are 0.67 and 0.89 for connections to D2 and D1 neurons, respectively. However, the small size of the samples means that the range of possible values is also broad (95% credibility intervals: D2, [0.35, 0.88]; D1, [0.56, 0.98]). Fortunately, the study of Gittis et al. (2010) is based on a much larger sample resulting in narrower posteriors shown in Figure 6A. Thanks to these narrower posterior curves, it is possible to conclude that FS interneurons preferentially target D1 neurons. Indeed, when we integrate Δ_{FS→D1– FS→D2} (Fig. 6A, inset), we find the probability that FS interneurons prefer to connect to D1 neurons is >0.99.

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Figure 6.

Bayesian analysis of connection probabilities of FS interneurons onto D1 and D2 SPNs. A, Connection probabilities of FS interneurons connecting to D1 and D2 SPNs according to the data of Gittis et al. (2010) who set a maximum distance of 250 μm between neurons. Inset, Density function for the difference in probability of connection. B, Same as in A, according to the data of Planert et al. (2010) who used a maximum distance of 100 μm instead. C, Posterior density functions for the decay rate of probability of connection for FS → D1 pairs assuming equiprobable sampling of neurons. D, Probabilities of connection given distance for three different values of β corresponding to the MAP estimates of each study and the intersection of the two posterior curves. E, F, Same for FS →D2 pairs as in C and D, respectively. Because the MAP estimate for Planert et al. (2010) coincides with the intersection of the two posteriors, only two exponential decays are tested in F.

The two studies seem to disagree as Planert et al. (2010) gives much higher estimates of p, but this is resolved by taking into account the maximum distance used by the two studies: 100 μm for Planert et al. (2010) and 250 μm for Gittis et al. (2010). Indeed, if we convert the posteriors for p into posteriors for the exponential decay rate β of the probability of connection given distance, with the assumption of equiprobable sampling as previously explained, we obtain posteriors that very largely overlap (Fig. 6C,E; Table 3), thus reconciling the two studies. In line with the already discussed overall smaller rate of connection to D2 neurons, the probability of connection drops much faster as distance increases for connections to D2 neurons (dropping to 50% after ∼100 μm; see Fig. 6F) than for connections to D1 neurons (50% connection rates occurring at a distance of at least 200 μm; see Fig. 6D). Consequently, there are at least two length-scales in the striatal microcircuit, with connections between SPNs falling to 50% probability within a few tens of micrometers (Fig. 4A,B), but connections to SPNs from FS interneurons falling to 50% probability at ≥100 μm (Fig. 6D,F).

PLTS interneurons make few local connections in striatum

We turn now to the connections that FS interneurons make on other interneurons of the striatum. To assess these, we analyzed data from Gittis et al. (2010) on connections FS interneurons make to PLTS, cholinergic, and other FS interneurons (Fig. 7A). Their data on connections between FS and PLTS interneurons were pooled with the data on the same connections from Szydlowski et al. (2013): We checked that the data from the two studies were in agreement by calculating posteriors separately for each study and found the density functions for the difference between the posteriors (Δ) for both directions (FS →PLTS and PLTS →FS) included 0 among their most likely values (results not shown).

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Figure 7.

Bayesian analysis of connection probabilities between striatal interneurons using a uniform prior. A, Posterior density functions for FS interneuron connections onto other interneurons according to the data of Gittis et al. (2010). B, Posterior density functions for the decay rate of probability of connection for FS → FS pairs assuming equiprobable sampling of neurons. Inset, Exponential decay function for the probability of connection between pairs of FS interneurons corresponding to the MAP estimate of the decay rate. C, Posterior density functions for PLTS interneuron connections onto other interneurons according to the data of Gittis et al. (2010). D, Posterior density functions for connections between cholinergic and TH interneurons according to data from Dorst et al. (2020). E, Posterior density functions for connections between cholinergic interneurons, NGF interneurons, and SPNs according to the data of English et al. (2011) and Ibáñez-Sandoval et al. (2011). F, Posterior density functions for the decay rate of the probability of connection for NGF → SPN pairs assuming equiprobable sampling of neurons. Inset, Exponential decay function for the probability of connection between NGF → SPN pairs corresponding to the MAP estimate of the decay rate in F.

We see from these data that the probability of connection from FS interneurons to PLTS interneurons is low (p̂MAP=0.10), but uncertainty regarding these connections is quite large (95% credibility interval = [0.03, 0.29]), while the probability of connection to cholinergic interneurons is even more uncertain with a credibility interval ranging from 0 to 0.60 and therefore requires more investigation. Connections between FS interneurons are relatively common (p̂MAP=0.58) but with broad uncertainty (95% credibility interval = [0.32, 0.81]). While this broad uncertainty in p translates into a broad uncertainty for the decay rate β of the probability of connection given the distance between a pair of FS interneurons (Fig. 7B), we see that the distance dependence for pairs of FS interneurons is similar to that for connections of FS interneurons to SPNs with a half-distance for connection probability as a function of distance of ∼200 μm for β̂MAP=0.003μm−1. Unfortunately, Szydlowski et al. (2013) do not provide a maximum distance which prevents us from transforming f_p(p) for FS → PLTS pairs into the corresponding f_β(β).

The combined data of Gittis et al. (2010) and Szydlowski et al. (2013) (Fig. 7C) show no evidence that PLTS interneurons connect locally to other interneurons (p̂MAP=0 for all pairs), but connections to SPNs, although sparse, are clearly established within a maximum distance of 250 μm (p̂MAP=0.033, 95% credibility interval = [0.010, 0.114]). Compared with FS interneurons, there is less uncertainty concerning the rates of connection for these PLTS interneurons, as evidenced by the smaller credibility intervals shown in Figure 7C. It remains to be seen whether there is an asymmetry in connection probability to D1 and D2 SPNs as the experimenters did not make this distinction when testing PLTS → SPN connections.

Further evidence that the effect of cholinergic interneurons onto SPNs is mediated by GABA interneurons

A recent study (Dorst et al., 2020) reported intracellular recording data on connections between cholinergic interneurons and the subtype of GABAergic interneurons that express TH (Table 1). When we apply our Bayesian method to this dataset (Fig. 7D), we find that TH and cholinergic interneurons connect reciprocally to one another quite frequently and with practically equal probabilities (p̂Ach→TH=0.268,p̂TH→Ach=0.260), with uncertainty estimates, which are relatively good compared with other interneuron connections (95% credibility interval = [0.159, 0.396] for TH → ACh connections, and [0.157, 0.420] for ACh → TH connections).

The activity of cholinergic interneurons indirectly affects SPNs via at least one type of GABAergic interneuron (English et al., 2011). To examine this route, we combine pairwise intracellular recording data from English et al. (2011) on connections between cholinergic interneurons and NPY-NGF interneurons, with data from Ibáñez-Sandoval et al. (2011) on connections from NPY-NGF interneurons to SPNs. Our analysis (Fig. 7E) reveals that cholinergic neurons connect frequently to NPY-NGF interneurons, which in turn connect very frequently to SPNs, making them an effective relay of cholinergic signals; this relay may also be regulated by the NPY-NGF interneurons frequent feedback connections on cholinergic interneurons. Furthermore, given that Ibáñez-Sandoval et al. (2011) used a maximum distance between neurons of 100 μm, it is possible to transform the posteriors for the probability of NGF → SPN into posteriors for the exponential decay rate β of probability of connection given distance. The MAP of β is the lowest we have found at ∼0.002 μm^–1 (Fig. 7F; Table 3). This means that, even at a distance of 100 μm from an NGF interneuron, an SPN still has a 0.8 probability of receiving a connection from this interneuron, which partly explains the effectiveness of the cholinergic system in regulating SPN activity.

Evidence used to compare SPN subtype connection rates in WT and Huntington's disease mice is insufficient

To this point, we have used our Bayesian approach to evaluate the probability of connection, the evidence for it, and (where possible) its dependence on the distance between neurons for every unique connection within the striatal microcircuit (for which there are extant data). We turn now to showing how our Bayesian approach lets us not just construct a map of the microcircuit, but also quantitatively test evidence for changes in the microcircuit. To do so, in this section, we evaluate evidence that connections between SPNs change in a mouse model of Huntington's disease (Cepeda et al., 2013); in the next section, we evaluate evidence for how connections between SPNs change over development.

The study of Cepeda et al. (2013) used smaller samples of identified SPN pairs than the (Taverna et al., 2008) and (Planert et al., 2010) studies of SPN connectivity (Table 4); consequently, the resulting posteriors are notably impacted by the choice of the prior (Fig. 8A,D). Indeed, the posterior curves obtained from the data of Cepeda et al. (2013) with a uniform prior or the prior based on previous literature look very different, contrary to those of Taverna et al. (2008) (compare Fig. 1C and Fig. 1G) and Planert et al. (2010) (compare Fig. 1D and Fig. 1H). In particular, the posteriors for WT mice obtained from the prior based on the previous literature look very similar to the initial prior (Fig. 8D) simply because of the small number of samples. Independently of the choice of prior, the posteriors overlap so much that it is not possible to confirm the rates of connection differ between any pairs of SPNs in the WT mice (Fig. 8A,D). Given how broad the posteriors are, we infer that this lack of difference is because of insufficient data rather than a true absence of difference.

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Figure 8.

Bayesian analysis of the study by Cepeda et al. (2013), comparing the probability of lateral SPN connections in WT mice and a model of Huntington's disease. A, Posterior density functions for the probabilities of connection in the WT mice using a uniform prior. Bars underneath represent the 95% credibility intervals. The curves for D2 → D1 and D1 → D2 coincide exactly. B, Posterior density functions for Huntington's disease animals using a uniform prior. The curves for D2 → D1 and D1 → D2 also coincide exactly. C, Probability density function for the difference in probabilities of connection for D1 → D1 pairs between the two animal groups. D-F, Same as in A-C using the prior based on the past literature.

Crucially, this lack of data is also true when comparing connection rates between the WT and Huntington's model mice (Fig. 8A,B or Fig. 8D,E). We find no evidence to support one of the conclusions reached by the authors that D1 → D1 connections are more likely in the Huntington's model than WT mice. Indeed, if we plot the density functions for the difference in probabilities of connection for D1 → D1 pairs between the two animal groups, we can see that it is very probable for this difference to be 0, but also any other value ranging from ∼−0.3 to 0.3 if we use a uniform prior (Fig. 8C) or a slightly more conservative −0.2 to 0.15 using the prior based on previous literature (Fig. 8F).

D2 SPN connection asymmetries appear during development

The development of connections between SPNs and their asymmetry can be tracked through postnatal development thanks to a recent study by Krajeski et al. (2019) who measured the probability of connection for different SPN pairs at three different stages of postnatal mouse development. The researchers reported that D1 neurons established lateral connections earlier than D2 neurons (a reproduction of these results with our added Wilson CIs is shown in Fig. 9A–C). We used our Bayesian approach to check this conclusion against the uncertainty in the experimental data, and provide further details of the development of the striatal microcircuit.

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Figure 9.

Postnatal development of the lateral connections of SPNs using data from Krajeski et al. (2019). A-C, Point estimates of the probabilities of connection at different developmental stages from Krajeski et al. (2019). We add here the 95% Wilson CIs. D-F, Posterior probability density functions for the probability of connections between SPNs at each developmental stage. Colored bars underneath the plot represent the 95% credibility intervals. A uniform prior as in Figure 1C is used. Inset, Density function for the difference in probability of connection for pairs with a D2 presynaptic neuron. G, H, Density functions for the difference in connection probabilities for each pair of neuron types between consecutive stages of postnatal development.

In order to apply our Bayesian method to these data, we resort to the uniform prior, since we have no particular expectation about these connection rates at these stages of development, and obtain the posteriors for each combination of neurons shown in Figure 9D–F. We have also tested the Jeffreys prior and obtained practically identical results (not shown). We can see that D1 neurons have already made some connections in the first 3-6 d of postnatal development (P3-P6; Fig. 9D), but it is hard to say from inspecting the posteriors at each subsequent developmental stage (P9-P12 and P21-P28) whether the connections made by D1 SPNs continue to develop or have already finished by P3-P6 (Fig. 9E,F). If we instead look at the difference in posteriors (f_Δ) between consecutive developmental stages (Fig. 9G,H), we see some evidence that connections made by D1 neurons continue to develop, with respective probabilities of 0.81 (P3-P6 to P9-P12) and 0.77 (P9-P12 to P21-P28) that the connection density of D1 neurons increases (probabilities again found by integrating f_Δ between 0 and 1). Comparing the earliest (P3-P6) and latest (P21-P28) stages gave a similar probability of 0.85 that D1 connections increased (data not shown).

For presynaptic D2 neurons on the other hand, it is clear that no or very few connections are present at P3-P6, and that they gradually appear later (Fig. 9D–F). Computing the difference in posteriors (f_Δ) for P(D2 → D2) and P(D2 → D1) between consecutive stages (Fig. 9G,H), we find the probability that D2 SPNs increase their connection density is 0.95 between P3-P6 and P9-P12 and 0.92 between P9-P12 and P21-P28, indicating a gradual development of these connections up to postnatal days 21-28. By also calculating fΔ(D2→D1)−(D2→D2) at each of these three stages (plotted as insets in Fig. 9D–F), we find good evidence that D2 neurons first connect to other D2 neurons before connecting to D1 neurons, supporting the claim by the authors of the original article (Krajeski et al., 2019). Notably, our analyses in this paper have thus shown that, while there is no evidence for a difference in the probability of presynaptic D2 SPNs connecting to D1 or D2 SPNs in the adult striatum (Fig. 2B,F), there is evidence that the D2 → D1 and D2 → D2 connections develop at different rates.

Our finding of strong evidence that D1 neurons are less likely to receive connections from SPNs than D2 neurons, both in adults (Fig. 2) and at later stages of development (Fig. 9), implies a role for active wiring processes in the developing striatum. We considered a simple model of a passive wiring process in which the contact of an axon from a first neuron onto the dendrite of a second neuron is determined only by the probability that an axon segment and a dendritic segment simultaneously occupy the same location (Liley and Wright, 1994; Kalisman et al., 2003; Humphries et al., 2010). For SPNs, we have the repeated observation that D1 SPNs have denser dendritic trees for the same volume as D2 SPNs (Gertler et al., 2008; Fujiyama et al., 2011; Gagnon et al., 2017). A passive wiring model would thus predict that D1 SPN dendrites receive more axonal contacts than D2 SPN dendrites from the same presynaptic type of SPN.

If this were true, we would expect to find in our analyses here that the asymmetry of connection rates would depend on the type of the postsynaptic neuron, when we instead find it depends on the type of presynaptic neuron; and we would expect D1 → D1 connections to be quite numerous when indeed these are quite rare. Together with our confirmation that the data of Krajeski et al. (2019) show D1 and D2 neurons develop their connections at a different rate, our analyses thus suggest that there is an active wiring process in striatal development that causes either an underexpression of connections to D1 SPNs or overexpression of connections to D2 SPNs.