Deep Learning for Computing Convergence Rates of Markov Chains

Convergence rate analysis for general state-space Markov chains is fundamentally important in areas such as Markov chain Monte Carlo and algorithmic analysis (for computing explicit convergence bounds). This problem, however, is notoriously difficult because traditional analytical methods often do not generate practically useful convergence bounds for realistic Markov chains. We propose the Deep Contractive Drift Calculator (DCDC), the first general-purpose sample-based algorithm for bounding the convergence of Markov chains to stationarity in Wasserstein distance. The DCDC has two components. First, inspired by the new convergence analysis framework in (Qu et.al, 2023), we introduce the Contractive Drift Equation (CDE), the solution of which leads to an explicit convergence bound. Second, we develop an efficient neural-network-based CDE solver. Equipped with these two components, DCDC solves the CDE and converts the solution into a convergence bound. We analyze the sample complexity of the algorithm and further demonstrate the effectiveness of the DCDC by generating convergence bounds for realistic Markov chains arising from stochastic processing networks as well as constant step-size stochastic optimization

Computable Bounds on Convergence of Markov Chains in Wasserstein Distance

We introduce a unified framework to estimate the convergence of Markov chains to equilibrium using Wasserstein distance. The framework provides convergence bounds with various rates, ranging from polynomial to exponential, all derived from a single contractive drift condition. This approach removes the need for finding a specific set with drift outside and contraction inside. The convergence bounds are explicit, as they can be estimated based on one-step expectations and do not rely on equilibrium-related quantities. To enhance the applicability of the framework, we introduce the large M technique and the boundary removal technique. We illustrate these methods in queueing models and algorithms in stochastic optimization

A POMC-originated circuit regulates stress-induced hypophagia, depression, and anhedonia

Chronic stress causes dysregulations of mood and energy homeostasis, but the neurocircuitry underlying these alterations remain to be fully elucidated. Here we demonstrate that chronic restraint stress in mice results in hyperactivity of pro-opiomelanocortin neurons in the arcuate nucleus of the hypothalamus (POMCARH neurons) associated with decreased neural activities of dopamine neurons in the ventral tegmental area (DAVTA neurons). We further revealed that POMCARH neurons project to the VTA and provide an inhibitory tone to DAVTA neurons via both direct and indirect neurotransmissions. Finally, we show that photoinhibition of the POMCARH→VTA circuit in mice increases body weight and food intake, and reduces depression-like behaviors and anhedonia in mice exposed to chronic restraint stress. Thus, our results identified a novel neurocircuitry regulating feeding and mood in response to stress

17β-estradiol promotes acute refeeding in hungry mice via membrane-initiated ERα signaling

Objective: Estrogen protects animals from obesity through estrogen receptor α (ERα), partially by inhibiting overeating in animals fed ad libitum. However, the effects of estrogen on feeding behavior in hungry animals remain unclear. In this study, we examined the roles of 17β-estradiol (E2) and ERα in the regulation of feeding in hungry female animals and explored the underlying mechanisms. Methods: Wild-type female mice with surgical depletion of endogenous estrogens were used to examine the effects of E2 supplementation on acute refeeding behavior after starvation. ERα-C451A mutant mice deficient in membrane-bound ERα activity and ERα-AF20 mutant mice lacking ERα transcriptional activity were used to further examine mechanisms underlying acute feeding triggered by either fasting or central glucopenia (induced by intracerebroventricular injections of 2-deoxy-D-glucose). We also used electrophysiology to explore the impact of these ERα mutations on the neural activities of ERα neurons in the hypothalamus. Results: In the wild-type female mice, ovariectomy reduced fasting-induced refeeding, which was restored by E2 supplementation. The ERα-C451A mutation, but not the ERα-AF20 mutation, attenuated acute feeding induced by either fasting or central glucopenia. The ERα-C451A mutation consistently impaired the neural responses of hypothalamic ERα neurons to hypoglycemia. Conclusion: In addition to previous evidence that estrogen reduces deviations in energy balance by inhibiting eating at a satiated state, our findings demonstrate the unexpected role of E2 that promotes eating in hungry mice, also contributing to the stability of energy homeostasis. This latter effect specifically requires membrane-bound ERα activity

Estrogen receptor-α expressing neurons in the ventrolateral VMH regulate glucose balance

Brain glucose-sensing neurons detect glucose fluctuations and prevent severe hypoglycemia, but mechanisms mediating functions of these glucose-sensing neurons are unclear. Here we report that estrogen receptor-α (ERα)-expressing neurons in the ventrolateral subdivision of the ventromedial hypothalamic nucleus (vlVMH) can sense glucose fluctuations, being glucose-inhibited neurons (GI-ERαvlVMH) or glucose-excited neurons (GE-ERαvlVMH). Hypoglycemia activates GI-ERαvlVMH neurons via the anoctamin 4 channel, and inhibits GE-ERαvlVMH neurons through opening the ATP-sensitive potassium channel. Further, we show that GI-ERαvlVMH neurons preferentially project to the medioposterior arcuate nucleus of the hypothalamus (mpARH) and GE-ERαvlVMH neurons preferentially project to the dorsal Raphe nuclei (DRN). Activation of ERαvlVMH to mpARH circuit and inhibition of ERαvlVMH to DRN circuit both increase blood glucose. Thus, our results indicate that ERαvlVMH neurons detect glucose fluctuations and prevent severe hypoglycemia in mice

Estrogen receptor-α expressing neurons in the ventrolateral VMH regulate glucose balance.

Brain glucose-sensing neurons detect glucose fluctuations and prevent severe hypoglycemia, but mechanisms mediating functions of these glucose-sensing neurons are unclear. Here we report that estrogen receptor-α (ERα)-expressing neurons in the ventrolateral subdivision of the ventromedial hypothalamic nucleus (vlVMH) can sense glucose fluctuations, being glucose-inhibited neurons (GI-ERαvlVMH) or glucose-excited neurons (GE-ERαvlVMH). Hypoglycemia activates GI-ERαvlVMH neurons via the anoctamin 4 channel, and inhibits GE-ERαvlVMH neurons through opening the ATP-sensitive potassium channel. Further, we show that GI-ERαvlVMH neurons preferentially project to the medioposterior arcuate nucleus of the hypothalamus (mpARH) and GE-ERαvlVMH neurons preferentially project to the dorsal Raphe nuclei (DRN). Activation of ERαvlVMH to mpARH circuit and inhibition of ERαvlVMH to DRN circuit both increase blood glucose. Thus, our results indicate that ERαvlVMH neurons detect glucose fluctuations and prevent severe hypoglycemia in mice

An estrogen-sensitive hypothalamus-midbrain neural circuit controls thermogenesis and physical activity

Estrogen receptor–α (ERα) expressed by neurons in the ventrolateral subdivision of the ventromedial hypothalamic nucleus (ERαvlVMH) regulates body weight in females, but the downstream neural circuits mediating this biology remain largely unknown. Here we identified a neural circuit mediating the metabolic effects of ERαvlVMH neurons. We found that selective activation of ERαvlVMH neurons stimulated brown adipose tissue (BAT) thermogenesis, physical activity, and core temperature and that ERαvlVMH neurons provide monosynaptic glutamatergic inputs to 5-hydroxytryptamine (5-HT) neurons in the dorsal raphe nucleus (DRN). Notably, the ERαvlVMH → DRN circuit responds to changes in ambient temperature and nutritional states. We further showed that 5-HTDRN neurons mediate the stimulatory effects of ERαvlVMH neurons on BAT thermogenesis and physical activity and that ERα expressed by DRN-projecting ERαvlVMH neurons is required for the maintenance of energy balance. Together, these findings support a model that ERαvlVMH neurons activate BAT thermogenesis and physical activity through stimulating 5-HTDRN neurons

An Estrogen-Sensitive Hypothalamus-Midbrain Neural Circuit Controls Thermogenesis and Physical Activity

Estrogen receptor–α (ERα) expressed by neurons in the ventrolateral subdivision of the ventromedial hypothalamic nucleus (ERαvlVMH) regulates body weight in females, but the downstream neural circuits mediating this biology remain largely unknown. Here we identified a neural circuit mediating the metabolic effects of ERαvlVMH neurons. We found that selective activation of ERαvlVMH neurons stimulated brown adipose tissue (BAT) thermogenesis, physical activity, and core temperature and that ERαvlVMH neurons provide monosynaptic glutamatergic inputs to 5-hydroxytryptamine (5-HT) neurons in the dorsal raphe nucleus (DRN). Notably, the ERαvlVMH → DRN circuit responds to changes in ambient temperature and nutritional states. We further showed that 5-HTDRN neurons mediate the stimulatory effects of ERαvlVMH neurons on BAT thermogenesis and physical activity and that ERα expressed by DRN-projecting ERαvlVMH neurons is required for the maintenance of energy balance. Together, these findings support a model that ERαvlVMH neurons activate BAT thermogenesis and physical activity through stimulating 5-HTDRN neurons

Gabra5 plays a sexually dimorphic role in POMC neuron activity and glucose balance

Pro-opiomelanocortin (POMC) neurons are important for the regulation of body weight and glucose balance. The inhibitory tone to POMC neurons is mediated primarily by the GABA receptors. However, the detailed mechanisms and functions of GABA receptors are not well understood. The α5 subunit of GABAA receptor, Gabra5, is reported to regulate feeding, and we found that Gabra5 is highly expressed in POMC neurons. To explore the function of Gabra5 in POMC neurons, we knocked down Gabra5 specifically from mature hypothalamic POMC neurons using the clustered regularly interspaced short palindromic repeats (CRISPR)-Cas9 strategy. This POMC-specific knock-down of Gabra5 did not affect body weight or food intake in either male or female mice. Interestingly, the loss of Gabra5 caused significant increases in the firing frequency and resting membrane potential, and a decrease in the amplitude of the miniature inhibitory postsynaptic current (mIPSC) in male POMC neurons. However, the loss of Gabra5 only modestly decreased the frequency of mIPSC in female POMC neurons. Consistently, POMC-specific knock-down of Gabra5 significantly improved glucose tolerance in male mice but not in female mice. These results revealed a sexually dimorphic role of Gabra5 in POMC neuron activity and glucose balance, independent of body weight control