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

Abstract

The striatum's complex microcircuit is made by connections within and between its D1- and D2-receptor expressing projection neurons and at least five species of interneuron. Precise knowledge of this circuit is likely essential to understanding striatum's functional roles and its dysfunction in a wide range of movement and cognitive disorders. We introduce here a Bayesian approach to mapping neuron connectivity using intracellular recording data, which lets us simultaneously evaluate the probability of connection between neuron types, the strength of evidence for it, and its dependence on distance. Using it to synthesize a complete map of the mouse striatum, we find strong evidence for two asymmetries: a selective asymmetry of projection neuron connections, with D2 neurons connecting twice as densely to other projection neurons than do D1 neurons, but neither subtype preferentially connecting to another; and a length-scale asymmetry, with interneuron connection probabilities remaining non-negligible at more than twice the distance of projection neuron connections. We further show that our Bayesian approach can evaluate evidence for wiring changes, using data from the developing striatum and a mouse model of Huntington's disease. By quantifying the uncertainty in our knowledge of the microcircuit, our approach reveals a wide range of potential striatal wiring diagrams consistent with current data.SIGNIFICANCE STATEMENT To properly understand a neuronal circuit's function, it is important to have an accurate picture of the rate of connection between individual neurons and how this rate changes with the distance separating pairs of neurons. We present a Bayesian method for extracting this information from experimental data and apply it to the mouse striatum, a subcortical structure involved in learning and decision-making, which is made up of a variety of different projection neurons and interneurons. Our resulting statistical map reveals not just the most robust estimates of the probability of connection between neuron types, but also the strength of evidence for them, and their dependence on distance.

Keywords: Bayesian inference; connectivity; interneurons; microcircuitry; spiny projection neurons; striatum.

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.

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).

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.

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.

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.

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.

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.

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.

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.

Figure 10.

Map of the striatum microcircuitry based on the MAP estimates for p and, when a maximum intersomatic distance was available, the decay rate β assuming equiprobable sampling. Line thickness is indicative of the relative probability of these connections. Connections between and within SPN subtypes are assumed to be the same for a given presynaptic subtype, as established in the main text, and the two different estimates for p correspond to the two different maximum distances used in Taverna et al. (2008) and Planert et al. (2010). Modelers wishing to use this map should be aware of the relative population size of these different neurons. For instance, although the probability of connection between SPNs is relatively small compared with connections from FS interneurons, this is potentially counterbalanced by the much greater number of SPNs within a given volume (Humphries et al., 2010). The map also necessarily omits known connections for which there are no appropriate intracellular recording data.