We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the FitzHugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons’ initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes place, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is a solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations or non-local partial differential equations resembling the McKean-Vlasov-Fokker-Planck equations. We prove the well-posedness of the McKean-Vlasov equations, i.e. the existence and uniqueness of a solution. We also show the results of some numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiments also indicate that the McKean-Vlasov-Fokker-Planck equations may be a good way to understand the mean-field dynamics through, e.g. a bifurcation analysis.Mathematics Subject Classification (2000): 60F99, 60B10, 92B20, 82C32, 82C80, 35Q80.

We study localised activity patterns in neural field equations posed on the Euclidean plane; such models are commonly used to describe the coarse-grained activity of large ensembles of cortical neurons in a spatially continuous way. We employ matrix-free Newton-Krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation. This opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent PDE formulation. We present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking. The regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function. The results presented form a basis for the general study of localised cortical activity with inputs and, more specifically, for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input.

How do the multiple cortico-basal ganglia-thalamo-cortical loops interact? Are they parallel and fully independent or controlled by an arbitrator, or are they hierarchically organized? We introduce here a set of four key concepts, integrated and evaluated by means of a neuro-computational model, that bring together current ideas regarding cortex-basal ganglia interactions in the context of habit learning. According to key concept 1, each loop learns to select an intermediate objective at a different abstraction level, moving from goals in the ventral striatum to motor in the putamen. Key concept 2 proposes that the cortex integrates the basal ganglia selection with environmental information regarding the achieved objective. Key concept 3 claims shortcuts between loops, and key concept 4 predicts that loops compute their own prediction error signal for learning. Computational benefits of the key concepts are demonstrated. Contrasting with former concepts of habit learning, the loops collaborate to select goal-directed actions while training slower shortcuts develops habitual responses.

Theories and models of the basal ganglia have mainly focused on the role of three different corticothalamic pathways: direct, indirect and hyperdirect. Although the indirect and the hyperdirect pathways are linked through the bidirectional connections between the subthalamic nucleus (STN) and the external globus pallidus (GPe), the role of their interactions has been mainly discussed in the context of a dysfunction (abnormal oscillations in Parkinson's disease) and not of its function. We here propose a novel role for the loop formed by the STN and the GPe. We show, through a neuro-computational model, that this loop can bias the selection of actions during the exploratory period after a change in the environmental conditions towards alternative responses. Testing well-known alternative solutions before completely random actions can reduce the time required for the search of a new response after a rule change. Our simulations further show that the knowledge acquired by the indirect pathway can be transferred into a stable memory via learning in the hyperdirect pathway to establish the blocking of unwanted responses. After a rule switch, first the indirect pathway learns to inhibit the previously correct actions. Once the new correct association is learned, the inhibition is transferred to the hyperdirect pathway through synaptic plasticity.

Previous computational model-based approaches for understanding the dynamic changes related to Parkinson's disease made particular assumptions about Parkinson's disease-related activity changes or specified dopamine-dependent activation or learning rules. Inspired by recent model-based analysis of resting-state fMRI, we have taken a data-driven approach. We fit the free parameters of a spiking neurocomputational model to match correlations of blood oxygen level-dependent signals between different basal ganglia nuclei and obtain subject-specific neuro-computational models of two subject groups: Parkinson patients and matched controls. When comparing mean firing rates at rest and connectivity strengths between the control and Parkinsonian model groups, several significant differences were found that are consistent with previous experimental observations. We discuss the implications of our approach and compare its results also with the popular "rate model" of the basal ganglia. Our study suggests that a model-based analysis of imaging data from healthy and Parkinsonian subjects is a promising approach for the future to better understand Parkinson-related changes in the basal ganglia and corresponding treatments. K E Y W O R D S BOLD correlations, data fitting, firing rate, spiking neuron model How to cite this article: Maith O, Villagrasa Escudero F, Dinkelbach HÜ, et al. A computational model-based analysis of basal ganglia pathway changes in Parkinson's disease inferred from resting-state fMRI.