The mechanics of correlated variability in segregated cortical excitatory subnetworks.

Understanding the genesis of shared trial-to-trial variability in neuronal population activity within the sensory cortex is critical to uncovering the biological basis of information processing in the brain. Shared variability is often a reflection of the structure of cortical connectivity since it likely arises, in part, from local circuit inputs. A series of experiments from segregated networks of (excitatory) pyramidal neurons in the mouse primary visual cortex challenge this view. Specifically, the across-network correlations were found to be larger than predicted given the known weak cross-network connectivity. We aim to uncover the circuit mechanisms responsible for these enhanced correlations through biologically motivated cortical circuit models. Our central finding is that coupling each excitatory subpopulation with a specific inhibitory subpopulation provides the most robust network-intrinsic solution in shaping these enhanced correlations. This result argues for the existence of excitatory-inhibitory functional assemblies in early sensory areas which mirror not just response properties but also connectivity between pyramidal cells. Furthermore, our findings provide theoretical support for recent experimental observations showing that cortical inhibition forms structural and functional subnetworks with excitatory cells, in contrast to the classical view that inhibition is a nonspecific blanket suppression of local excitation.

Investigating the ability of astrocytes to drive neural network synchrony.

Recent experimental works have implicated astrocytes as a significant cell type underlying several neuronal processes in the mammalian brain, from encoding sensory information to neurological disorders. Despite this progress, it is still unclear how astrocytes are communicating with and driving their neuronal neighbors. While previous computational modeling works have helped propose mechanisms responsible for driving these interactions, they have primarily focused on interactions at the synaptic level, with microscale models of calcium dynamics and neurotransmitter diffusion. Since it is computationally infeasible to include the intricate microscale details in a network-scale model, little computational work has been done to understand how astrocytes may be influencing spiking patterns and synchronization of large networks. We overcome this issue by first developing an "effective" astrocyte that can be easily implemented to already established network frameworks. We do this by showing that the astrocyte proximity to a synapse makes synaptic transmission faster, weaker, and less reliable. Thus, our "effective" astrocytes can be incorporated by considering heterogeneous synaptic time constants, which are parametrized only by the degree of astrocytic proximity at that synapse. We then apply our framework to large networks of exponential integrate-and-fire neurons with various spatial structures. Depending on key parameters, such as the number of synapses ensheathed and the strength of this ensheathment, we show that astrocytes can push the network to a synchronous state and exhibit spatially correlated patterns.

Revising Berg-Purcell for finite receptor kinetics.

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.

Receptor recharge time drastically reduces the number of captured particles.

Many diverse biological systems are described by randomly moving particles that can be captured by traps in their environment. Examples include neurotransmitters diffusing in the synaptic cleft before binding to receptors and prey roaming an environment before capture by predators. In most cases, the traps cannot capture particles continuously. Rather, each trap must wait a transitory "recharge" time after capturing a particle before additional captures. This recharge time is often overlooked. In the case of instant recharge, the average number of particles captured before they escape grows linearly in the total number of particles. In stark contrast, we prove that for any nonzero recharge time, the average number of captured particles grows at most logarithmically in the total particle number. This is a fundamental effect of recharge, as it holds under very general assumptions on particle motion and spatial domain. Furthermore, we characterize the parameter regime in which a given recharge time will dramatically affect a system, allowing researchers to easily verify if they need to account for recharge in their specific system. Finally, we consider a few examples, including a neural system in which recharge reduces neurotransmitter bindings by several orders of magnitude.

Mathematical investigation of IP3-dependent calcium dynamics in astrocytes.

We study evoked calcium dynamics in astrocytes, a major cell type in the mammalian brain. Experimental evidence has shown that such dynamics are highly variable between different trials, cells, and cell subcompartments. Here we present a qualitative analysis of a recent mathematical model of astrocyte calcium responses. We show how the major response types are generated in the model as a result of the underlying bifurcation structure. By varying key channel parameters, mimicking blockers used by experimentalists, we manipulate this underlying bifurcation structure and predict how the distributions of responses can change. We find that store-operated calcium channels, plasma membrane bound channels with little activity during calcium transients, have a surprisingly strong effect, underscoring the importance of considering these channels in both experiments and mathematical settings. Variation in the maximum flow in different calcium channels is also shown to determine the range of stable oscillations, as well as set the range of frequencies of the oscillations. Further, by conducting a randomized search through the parameter space and recording the resulting calcium responses, we create a database that can be used by experimentalists to help estimate the underlying channel distribution of their cells.

The logic of recurrent circuits in the primary visual cortex.

Recurrent cortical activity sculpts visual perception by refining, amplifying or suppressing visual input. However, the rules that govern the influence of recurrent activity remain enigmatic. We used ensemble-specific two-photon optogenetics in the mouse visual cortex to isolate the impact of recurrent activity from external visual input. We found that the spatial arrangement and the visual feature preference of the stimulated ensemble and the neighboring neurons jointly determine the net effect of recurrent activity. Photoactivation of these ensembles drives suppression in all cells beyond 30 µm but uniformly drives activation in closer similarly tuned cells. In nonsimilarly tuned cells, compact, cotuned ensembles drive net suppression, while diffuse, cotuned ensembles drive activation. Computational modeling suggests that highly local recurrent excitatory connectivity and selective convergence onto inhibitory neurons explain these effects. Our findings reveal a straightforward logic in which space and feature preference of cortical ensembles determine their impact on local recurrent activity.

The mechanics of correlated variability in segregated cortical excitatory subnetworks.

Understanding the genesis of shared trial-to-trial variability in neural activity within sensory cortex is critical to uncovering the biological basis of information processing in the brain. Shared variability is often a reflection of the structure of cortical connectivity since this variability likely arises, in part, from local circuit inputs. A series of experiments from segregated networks of (excitatory) pyramidal neurons in mouse primary visual cortex challenge this view. Specifically, the across-network correlations were found to be larger than predicted given the known weak cross-network connectivity. We aim to uncover the circuit mechanisms responsible for these enhanced correlations through biologically motivated cortical circuit models. Our central finding is that coupling each excitatory subpopulation with a specific inhibitory subpopulation provides the most robust network-intrinsic solution in shaping these enhanced correlations. This result argues for the existence of excitatory-inhibitory functional assemblies in early sensory areas which mirror not just response properties but also connectivity between pyramidal cells.

Cortical VIP neurons locally control the gain but globally control the coherence of gamma band rhythms.

Gamma band synchronization can facilitate local and long-range neural communication. In the primary visual cortex, visual stimulus properties within a specific location determine local synchronization strength, while the match of stimulus properties between distant locations controls long-range synchronization. The neural basis for the differential control of local and global gamma band synchronization is unknown. Combining electrophysiology, optogenetics, and computational modeling, we found that VIP disinhibitory interneurons in mouse cortex linearly scale gamma power locally without changing its stimulus tuning. Conversely, they suppress long-range synchronization when two regions process non-matched stimuli, tuning gamma coherence globally. Modeling shows that like-to-like connectivity across space and specific VIP→SST inhibition capture these opposing effects. VIP neurons thus differentially impact local and global properties of gamma rhythms depending on visual stimulus statistics. They may thereby construct gamma-band filters for spatially extended but continuous image features, such as contours, facilitating the downstream generation of coherent visual percepts.

Cell-type-specific plasticity of inhibitory interneurons in the rehabilitation of auditory cortex after peripheral damage.

Peripheral sensory organ damage leads to compensatory cortical plasticity that is associated with a remarkable recovery of cortical responses to sound. The precise mechanisms that explain how this plasticity is implemented and distributed over a diverse collection of excitatory and inhibitory cortical neurons remain unknown. After noise trauma and persistent peripheral deficits, we found recovered sound-evoked activity in mouse A1 excitatory principal neurons (PNs), parvalbumin- and vasoactive intestinal peptide-expressing neurons (PVs and VIPs), but reduced activity in somatostatin-expressing neurons (SOMs). This cell-type-specific recovery was also associated with cell-type-specific intrinsic plasticity. These findings, along with our computational modelling results, are consistent with the notion that PV plasticity contributes to PN stability, SOM plasticity allows for increased PN and PV activity, and VIP plasticity enables PN and PV recovery by inhibiting SOMs.

Extending the IP3 receptor model to include competition with partial agonists.

The inositol 1,4,5-trisphosphate (IP(3)) receptor is a Ca(2+) channel located in the endoplasmic reticulum and is regulated by IP(3) and Ca(2+). This channel is critical to calcium signaling in cell types as varied as neurons and pancreatic beta cells to mast cells. De Young and Keizer (1992) created an eight-state, nine-variable model of the IP(3) receptor. In their model, they accounted for three binding sites, a site for IP(3), activating Ca(2+), and deactivating Ca(2+). The receptor is only open if IP(3) and activating Ca(2+) is bound. Li and Rinzel followed up this paper in 1994 by introducing a reduction that made it into a two variable system. A recent publication by Rossi et al. (2009) studied the effect of introducing IP(3)-like molecules, referred to as partial agonists (PA), into the cell to determine the structure-function relationship between IP(3) and its receptor. Initial results suggest a competitive model, where IP(3) and PA fight for the same binding site. We extend the original eight-state model to a 12-state model in order to illustrate this competition, and perform a similar reduction to that of Li and Rinzel in the first modeling study we are aware of considering PA effect on an IP(3) receptor. Using this reduction we solve for the equilibrium open probability for calcium release in the model. We replicate graphs provided by the Rossi paper, and find that optimizing the subunit affinities for IP(3) and PA yields a good fit to the data. We plug our extended reduced model into a full cell model, in order to analyze the effects PA have on whole cell properties specifically the propagation of calcium waves in two dimensions. We conclude that PA creates qualitatively different calcium dynamics than would simply reducing IP(3), but that effectively PA can act as an IP(3) knockdown.

Recurrent network dynamics shape direction selectivity in primary auditory cortex.

Detecting the direction of frequency modulation (FM) is essential for vocal communication in both animals and humans. Direction-selective firing of neurons in the primary auditory cortex (A1) has been classically attributed to temporal offsets between feedforward excitatory and inhibitory inputs. However, it remains unclear how cortical recurrent circuitry contributes to this computation. Here, we used two-photon calcium imaging and whole-cell recordings in awake mice to demonstrate that direction selectivity is not caused by temporal offsets between synaptic currents, but by an asymmetry in total synaptic charge between preferred and non-preferred directions. Inactivation of cortical somatostatin-expressing interneurons (SOM cells) reduced direction selectivity, revealing its cortical contribution. Our theoretical models showed that charge asymmetry arises due to broad spatial topography of SOM cell-mediated inhibition which regulates signal amplification in strongly recurrent circuitry. Together, our findings reveal a major contribution of recurrent network dynamics in shaping cortical tuning to behaviorally relevant complex sounds.

Role of trap recharge time on the statistics of captured particles.

We consider n particles diffusing freely in a domain. The boundary contains absorbing escape regions, where the particles can escape, and traps, where the particles can be captured. Modeled after biological examples such as receptors in the synaptic cleft and ambush predators waiting for prey, these traps, or capture regions, must recharge between captures. We are interested in characterizing the time courses of the number of particles remaining in the domain, the number of cumulative captures, and the number of available capture regions. We find that under certain conditions, the number of cumulative captures increases linearly in time with a slope and duration determined explicitly by the recharge rate of the capture regions. This recharge rate also determines the mean and variance of the clearance time, defined as the time it takes for all particles to leave the domain. Further, we find that while a finite recharge rate will always result in a lower number of captured particles when compared to instantaneous recharging, it can either increase or decrease the amount of variability. Lastly, we extend the model to partially absorbing traps in order to investigate the dynamics of receptor activation within an idealized synaptic cleft. We find that the width of the domain controls the amount of time that these receptors are activated, while the number of receptors controls the amplitude of activation. Our mathematical results are derived from considering this system in several ways: as a full spatial diffusion process with recharging traps, as a continuous-time Markov process on a discrete state space, and as a system of ordinary differential equations in a mean-field approximation.

Diversity of Evoked Astrocyte Ca2+ Dynamics Quantified through Experimental Measurements and Mathematical Modeling.

Astrocytes are a major cell type in the mammalian brain. They are not electrically excitable, but generate prominent Ca2+ signals related to a wide variety of critical functions. The mechanisms driving these Ca2+ events remain incompletely understood. In this study, we integrate Ca2+ imaging, quantitative data analysis, and mechanistic computational modeling to study the spatial and temporal heterogeneity of cortical astrocyte Ca2+ transients evoked by focal application of ATP in mouse brain slices. Based on experimental results, we tune a single-compartment mathematical model of IP3-dependent Ca2+ responses in astrocytes and use that model to study response heterogeneity. Using information from the experimental data and the underlying bifurcation structure of our mathematical model, we categorize all astrocyte Ca2+ responses into four general types based on their temporal characteristics: Single-Peak, Multi-Peak, Plateau, and Long-Lasting responses. We find that the distribution of experimentally-recorded response types depends on the location within an astrocyte, with somatic responses dominated by Single-Peak (SP) responses and large and small processes generating more Multi-Peak responses. On the other hand, response kinetics differ more between cells and trials than with location within a given astrocyte. We use the computational model to elucidate possible sources of Ca2+ response variability: (1) temporal dynamics of IP3, and (2) relative flux rates through Ca2+ channels and pumps. Our model also predicts the effects of blocking Ca2+ channels/pumps; for example, blocking store-operated Ca2+ (SOC) channels in the model eliminates Plateau and Long-Lasting responses (consistent with previous experimental observations). Finally, we propose that observed differences in response type distributions between astrocyte somas and processes can be attributed to systematic differences in IP3 rise durations and Ca2+ flux rates.