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Biological neural networks are often modeled as systems of coupled, nonlinear, ordinary or partial differential equations. The number of differential equations used to model a network increases with the size of the network and the level of detail used to model individual neurons and synapses. As one scales up the size of the simulation, it becomes essential to utilize powerful computing platforms. While many tools exist that solve these equations numerically, they are often platform-specific. Further, there is a high barrier of entry to developing flexible platform-independent general-purpose code that supports hardware acceleration on modern computing architectures such as GPUs/TPUs and Distributed Platforms. TensorFlow is a Python-based open-source package designed for machine learning algorithms. However, it is also a scalable environment for a variety of computations, including solving differential equations using iterative algorithms such as Runge-Kutta methods. In this article and the accompanying tutorials, we present a simple exposition of numerical methods to solve ordinary differential equations using Python and TensorFlow. The tutorials consist of a series of Python notebooks that, over the course of five sessions, will lead novice programmers from writing programs to integrate simple one-dimensional ordinary differential equations using Python to solving a large system (1000’s of differential equations) of coupled conductance-based neurons using a highly parallelized and scalable framework. Embedded with the tutorial is a physiologically realistic implementation of a network in the insect olfactory system. This system, consisting of multiple neuron and synapse types, can serve as a template to simulate other networks.

The antennal lobe, the insect equivalent of the olfactory bulb in mammals, is a dense network of excitatory projection neurons (PNs) and inhibitory local interneurons (LNs). In the locust, this network is known to generate elaborate spatiotemporal patterns of activity in response to an odor. Computational models of the antennal lobe have typically assumed that the input to the inhibitory interneurons is narrowly tuned. That is, each interneuron receives input from a small fraction of olfactory receptor neurons (ORNs) and responds reliably to only a small subset of odor inputs. However, we know from a number of experiments, this is not the case. Interneurons arborise extensively across the antennal lobe and receive input from nearly 70% of all ORNs in the antennae. Therefore, odor input to different LNs is not as distinguishable as one would expect if the neurons were narrowly tuned. Using a realistic computational model of the locust antennal lobe, we sought to understand the effect that broad odor tuning has on the repertoire of spatiotemporal patterns the network can reliably generate. We modeled PNs and LNs as conductance-based neurons with properties matching in vivo observations. Spiking ORN input drove PNs and LNs in a manner that replicated experimentally observed statistics of PN responses. Earlier studies have shown that competitive interactions between interneurons can result in distinct patterns of activity; spatiotemporal attractors that are defined by the topology of the inhibitory sub-network. The inhibitory network can, in turn, entrain projection neurons. Here, we explore the hypothesis that excitatory input from projection neurons act as perturbations that drive the inhibitory network through a sequence of attractors. We examined the dynamics of switching as a function of the balance between excitatory (PN and ORN drive to the interneurons) and inhibitory (reciprocal interactions between interneurons) drive in the LN network.

Brief AbstractWe analysed SARS-CoV-2 surveillance and contact tracing data from Karnataka, India up to 21 July 2020. We estimated metrics of infectiousness and the tendency for superspreading (overdispersion), and evaluated potential determinants of infectiousness and symptomaticity in COVID-19 cases. Among 956 cases confirmed to be forward-traced, 8.7% of index cases had 14.4% of contacts but caused 80% of all secondary cases, suggesting significant heterogeneity in individual-level transmissibility of SARS-CoV-2 which could not be explained by the degree of heterogeneity in underlying number of contacts. Secondary attack rate was 3.6% among 16715 close contacts. Transmission was higher when index case was aged >18 years, or was symptomatic (adjusted risk ratio, aRR 3.63), or was lab-confirmed ≥4 days after symptom onset (aRR 3.01). Probability of symptomatic infection increased with age, and symptomatic infectors were 8.16 times more likely to generate symptomatic secondaries. This could potentially cause a snowballing effect on infectiousness and clinical severity across transmission generations; further studies are suggested to confirm this. Mean serial interval was 5.4 days. Adding backward contact tracing and targeting control measures to curb super-spreading may be prudent. Due to low symptomaticity and infectivity, interventions aimed at children might have a relatively small impact on reducing transmission.Structured AbstractBackgroundIndia has experienced the second largest outbreak of COVID-19 globally, yet there is a paucity of studies analysing contact tracing data in the region. Such studies can elucidate essential transmission metrics which can help optimize disease control policies.MethodsWe analysed contact tracing data collected under the Integrated Disease Surveillance Programme from Karnataka, India between 9 March and 21 July 2020. We estimated metrics of disease transmission including the reproduction number (R), overdispersion (k), secondary attack rate (SAR), and serial interval. R and k were jointly estimated using a Bayesian Markov Chain Monte Carlo approach. We evaluated the effect of age and other factors on the risk of transmitting the infection, probability of asymptomatic infection, and mortality due to COVID-19.FindingsUp to 21 July, we found 111 index cases that crossed the super-spreading threshold of ≥8 secondary cases. R and k were most reliably estimated at R 0.75 (95% CI, 0.62-0.91) and k 0.12 (0.11-0.15) for confirmed traced cases (n=956); and R 0.91 (0.72-1.15) and k 0.22 (0.17-0.27) from the three largest clusters (n=394). Among 956 confirmed traced cases, 8.7% of index cases had 14.4% of contacts but caused 80% of all secondary cases. Among 16715 contacts, overall SAR was 3.6% (3.4-3.9) and symptomatic cases were more infectious than asymptomatic cases (SAR 7.7% vs 2.0%; aRR 3.63 [3.04-4.34]). As compared to infectors aged 19-44 years, children were less infectious (aRR 0.21 [0.07-0.66] for 0-5 years and 0.47 [0.32-0.68] for 6-18 years). Infectors who were confirmed ≥4 days after symptom onset were associated with higher infectiousness (aRR 3.01 [2.11-4.31]). Probability of symptomatic infection increased with age, and symptomatic infectors were 8.16 (3.29-20.24) times more likely to generate symptomatic secondaries. Serial interval had a mean of 5.4 (4.4-6.4) days with a Weibull distribution. Overall case fatality rate was 2.5% (2.4-2.7) which increased with age.ConclusionWe found significant heterogeneity in the individual-level transmissibility of SARS-CoV-2 which could not be explained by the degree of heterogeneity in the underlying number of contacts. To strengthen contact tracing in over-dispersed outbreaks, testing and tracing delays should be minimised, retrospective contact tracing should be considered, and contact tracing performance metrics should be utilised. Targeted measures to reduce potential superspreading events should be implemented. Interventions aimed at children might have a relatively small impact on reducing SARS-CoV-2 transmission owing to their low symptomaticity and infectivity. There is some evidence that symptomatic cases produce secondary cases that are more likely to be symptomatic themselves which may potentially cause a snowballing effect on infectiousness and clinical severity across transmission generations; further studies are needed to confirm this finding.FundingGiridhara R Babu is funded by an Intermediate Fellowship by the Wellcome Trust DBT India Alliance (Clinical and Public Health Research Fellowship); grant number: IA/CPHI/14/1/501499.