A spinal cord neuroprosthesis for locomotor deficits due to Parkinson's disease.

People with late-stage Parkinson's disease (PD) often suffer from debilitating locomotor deficits that are resistant to currently available therapies. To alleviate these deficits, we developed a neuroprosthesis operating in closed loop that targets the dorsal root entry zones innervating lumbosacral segments to reproduce the natural spatiotemporal activation of the lumbosacral spinal cord during walking. We first developed this neuroprosthesis in a non-human primate model that replicates locomotor deficits due to PD. This neuroprosthesis not only alleviated locomotor deficits but also restored skilled walking in this model. We then implanted the neuroprosthesis in a 62-year-old male with a 30-year history of PD who presented with severe gait impairments and frequent falls that were medically refractory to currently available therapies. We found that the neuroprosthesis interacted synergistically with deep brain stimulation of the subthalamic nucleus and dopaminergic replacement therapies to alleviate asymmetry and promote longer steps, improve balance and reduce freezing of gait. This neuroprosthesis opens new perspectives to reduce the severity of locomotor deficits in people with PD.

Walking naturally after spinal cord injury using a brain-spine interface.

A spinal cord injury interrupts the communication between the brain and the region of the spinal cord that produces walking, leading to paralysis1,2. Here, we restored this communication with a digital bridge between the brain and spinal cord that enabled an individual with chronic tetraplegia to stand and walk naturally in community settings. This brain-spine interface (BSI) consists of fully implanted recording and stimulation systems that establish a direct link between cortical signals3 and the analogue modulation of epidural electrical stimulation targeting the spinal cord regions involved in the production of walking4-6. A highly reliable BSI is calibrated within a few minutes. This reliability has remained stable over one year, including during independent use at home. The participant reports that the BSI enables natural control over the movements of his legs to stand, walk, climb stairs and even traverse complex terrains. Moreover, neurorehabilitation supported by the BSI improved neurological recovery. The participant regained the ability to walk with crutches overground even when the BSI was switched off. This digital bridge establishes a framework to restore natural control of movement after paralysis.

A framework for macroscopic phase-resetting curves for generalised spiking neural networks.

Brain rhythms emerge from synchronization among interconnected spiking neurons. Key properties of such rhythms can be gleaned from the phase-resetting curve (PRC). Inferring the PRC and developing a systematic phase reduction theory for large-scale brain rhythms remains an outstanding challenge. Here we present a theoretical framework and methodology to compute the PRC of generic spiking networks with emergent collective oscillations. We adopt a renewal approach where neurons are described by the time since their last action potential, a description that can reproduce the dynamical feature of many cell types. For a sufficiently large number of neurons, the network dynamics are well captured by a continuity equation known as the refractory density equation. We develop an adjoint method for this equation giving a semi-analytical expression of the infinitesimal PRC. We confirm the validity of our framework for specific examples of neural networks. Our theoretical framework can link key biological properties at the individual neuron scale and the macroscopic oscillatory network properties. Beyond spiking networks, the approach is applicable to a broad class of systems that can be described by renewal processes.

Implanted System for Orthostatic Hypotension in Multiple-System Atrophy.

Orthostatic hypotension is a cardinal feature of multiple-system atrophy. The upright posture provokes syncopal episodes that prevent patients from standing and walking for more than brief periods. We implanted a system to restore regulation of blood pressure and enable a patient with multiple-system atrophy to stand and walk after having lost these abilities because of orthostatic hypotension. This system involved epidural electrical stimulation delivered over the thoracic spinal cord with accelerometers that detected changes in body position. (Funded by the Defitech Foundation.).

Phase response approaches to neural activity models with distributed delay.

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

Macroscopic phase resetting-curves determine oscillatory coherence and signal transfer in inter-coupled neural circuits.

Macroscopic oscillations of different brain regions show multiple phase relationships that are persistent across time and have been implicated in routing information. While multiple cellular mechanisms influence the network oscillatory dynamics and structure the macroscopic firing motifs, one of the key questions is to identify the biophysical neuronal and synaptic properties that permit such motifs to arise. A second important issue is how the different neural activity coherence states determine the communication between the neural circuits. Here we analyse the emergence of phase-locking within bidirectionally delayed-coupled spiking circuits in which global gamma band oscillations arise from synaptic coupling among largely excitable neurons. We consider both the interneuronal (ING) and the pyramidal-interneuronal (PING) population gamma rhythms and the inter coupling targeting the pyramidal or the inhibitory neurons. Using a mean-field approach together with an exact reduction method, we reduce each spiking network to a low dimensional nonlinear system and derive the macroscopic phase resetting-curves (mPRCs) that determine how the phase of the global oscillation responds to incoming perturbations. This is made possible by the use of the quadratic integrate-and-fire model together with a Lorentzian distribution of the bias current. Depending on the type of gamma (PING vs. ING), we show that incoming excitatory inputs can either speed up the macroscopic oscillation (phase advance; type I PRC) or induce both a phase advance and a delay (type II PRC). From there we determine the structure of macroscopic coherence states (phase-locking) of two weakly synaptically-coupled networks. To do so we derive a phase equation for the coupled system which links the synaptic mechanisms to the coherence states of the system. We show that a synaptic transmission delay is a necessary condition for symmetry breaking, i.e. a non-symmetric phase lag between the macroscopic oscillations. This potentially provides an explanation to the experimentally observed variety of gamma phase-locking modes. Our analysis further shows that symmetry-broken coherence states can lead to a preferred direction of signal transfer between the oscillatory networks where this directionality also depends on the timing of the signal. Hence we suggest a causal theory for oscillatory modulation of functional connectivity between cortical circuits.

A stochastic-field description of finite-size spiking neural networks.

Neural network dynamics are governed by the interaction of spiking neurons. Stochastic aspects of single-neuron dynamics propagate up to the network level and shape the dynamical and informational properties of the population. Mean-field models of population activity disregard the finite-size stochastic fluctuations of network dynamics and thus offer a deterministic description of the system. Here, we derive a stochastic partial differential equation (SPDE) describing the temporal evolution of the finite-size refractory density, which represents the proportion of neurons in a given refractory state at any given time. The population activity-the density of active neurons per unit time-is easily extracted from this refractory density. The SPDE includes finite-size effects through a two-dimensional Gaussian white noise that acts both in time and along the refractory dimension. For an infinite number of neurons the standard mean-field theory is recovered. A discretization of the SPDE along its characteristic curves allows direct simulations of the activity of large but finite spiking networks; this constitutes the main advantage of our approach. Linearizing the SPDE with respect to the deterministic asynchronous state allows the theoretical investigation of finite-size activity fluctuations. In particular, analytical expressions for the power spectrum and autocorrelation of activity fluctuations are obtained. Moreover, our approach can be adapted to incorporate multiple interacting populations and quasi-renewal single-neuron dynamics.

Is There a Nonadditive Interaction Between Spontaneous and Evoked Activity? Phase-Dependence and Its Relation to the Temporal Structure of Scale-Free Brain Activity.

The aim of our study was to use functional magnetic resonance imaging to investigate how spontaneous activity interacts with evoked activity, as well as how the temporal structure of spontaneous activity, that is, long-range temporal correlations, relate to this interaction. Using an extremely sparse event-related design (intertrial intervals: 52-60 s), a novel blood oxygen level-dependent signal correction approach (accounting for spontaneous fluctuations using pseudotrials) and phase analysis, we provided direct evidence for a nonadditive interaction between spontaneous and evoked activity. We demonstrated the discrepancy between the present and previous observations on why a linear superposition between spontaneous and evoked activity can be seen by using co-occurring signals from homologous brain regions. Importantly, we further demonstrated that the nonadditive interaction can be characterized by phase-dependent effects of spontaneous activity, which is closely related to the degree of long-range temporal correlations in spontaneous activity as indexed by both power-law exponent and phase-amplitude coupling. Our findings not only contribute to the understanding of spontaneous brain activity and its scale-free properties, but also bear important implications for our understanding of neural activity in general.

Macroscopic phase-resetting curves for spiking neural networks.

The study of brain rhythms is an open-ended, and challenging, subject of interest in neuroscience. One of the best tools for the understanding of oscillations at the single neuron level is the phase-resetting curve (PRC). Synchronization in networks of neurons, effects of noise on the rhythms, effects of transient stimuli on the ongoing rhythmic activity, and many other features can be understood by the PRC. However, most macroscopic brain rhythms are generated by large populations of neurons, and so far it has been unclear how the PRC formulation can be extended to these more common rhythms. In this paper, we describe a framework to determine a macroscopic PRC (mPRC) for a network of spiking excitatory and inhibitory neurons that generate a macroscopic rhythm. We take advantage of a thermodynamic approach combined with a reduction method to simplify the network description to a small number of ordinary differential equations. From this simplified but exact reduction, we can compute the mPRC via the standard adjoint method. Our theoretical findings are illustrated with and supported by numerical simulations of the full spiking network. Notably our mPRC framework allows us to predict the difference between effects of transient inputs to the excitatory versus the inhibitory neurons in the network.

Theoretical connections between mathematical neuronal models corresponding to different expressions of noise.

Identifying the right tools to express the stochastic aspects of neural activity has proven to be one of the biggest challenges in computational neuroscience. Even if there is no definitive answer to this issue, the most common procedure to express this randomness is the use of stochastic models. In accordance with the origin of variability, the sources of randomness are classified as intrinsic or extrinsic and give rise to distinct mathematical frameworks to track down the dynamics of the cell. While the external variability is generally treated by the use of a Wiener process in models such as the Integrate-and-Fire model, the internal variability is mostly expressed via a random firing process. In this paper, we investigate how those distinct expressions of variability can be related. To do so, we examine the probability density functions to the corresponding stochastic models and investigate in what way they can be mapped one to another via integral transforms. Our theoretical findings offer a new insight view into the particular categories of variability and it confirms that, despite their contrasting nature, the mathematical formalization of internal and external variability is strikingly similar.

TEST-RETEST RELIABILITY OF TWO CLINICAL TESTS FOR THE ASSESSMENT OF HIP ABDUCTOR ENDURANCE IN HEALTHY FEMALES.

BACKGROUND: Substantial deficits in performance of hip abductor in patients with common lower extremity injuries are reported in literature. Therefore, assessing hip abductor endurance might be of major importance for clinicians and researchers. PURPOSES: The purpose of this study was to examine the test-retest reliability of two hip abductor endurance tests in healthy females. Learning effect, systematic difference in the rate of perceived exertion and relationship between endurance performance and some clinical characteristics of participants were also investigated. DESIGN: Observational study, with a test-retest design. METHODS: Thirty-six healthy females, aged 18-30 years, were recruited. In two identical assessment sessions, the participants performed an isometric hip abductor strength test and two different hip abductor endurance tests. RESULTS: Isometric and dynamic endurance tests demonstrated good test-retest reliability (intraclass correlation coefficients (ICC) = 0.73 and 0.78, respectively). The standard errors of measurement (SEM) and the minimal detectable changes (MDC) were, respectively, 19.8 and 54.9 seconds for isometric endurance test and 21.2 and 58.7 repetitions for dynamic endurance test. Moderate correlation between both endurance tests (r = 0.60, p = 0.0001) and weak correlation between dynamic endurance test and strength (r = 0.44, p = 0.008) were found. CONCLUSIONS: The results of the present study demonstrate good test-retest reliability of two non-instrumented clinical tests of hip abductor endurance in healthy females. LEVEL OF EVIDENCE: 2b.

A stochastic model of input effectiveness during irregular gamma rhythms.

Gamma-band synchronization has been linked to attention and communication between brain regions, yet the underlying dynamical mechanisms are still unclear. How does the timing and amplitude of inputs to cells that generate an endogenously noisy gamma rhythm affect the network activity and rhythm? How does such "communication through coherence" (CTC) survive in the face of rhythm and input variability? We present a stochastic modelling approach to this question that yields a very fast computation of the effectiveness of inputs to cells involved in gamma rhythms. Our work is partly motivated by recent optogenetic experiments (Cardin et al. Nature, 459(7247), 663-667 2009) that tested the gamma phase-dependence of network responses by first stabilizing the rhythm with periodic light pulses to the interneurons (I). Our computationally efficient model E-I network of stochastic two-state neurons exhibits finite-size fluctuations. Using the Hilbert transform and Kuramoto index, we study how the stochastic phase of its gamma rhythm is entrained by external pulses. We then compute how this rhythmic inhibition controls the effectiveness of external input onto pyramidal (E) cells, and how variability shapes the window of firing opportunity. For transferring the time variations of an external input to the E cells, we find a tradeoff between the phase selectivity and depth of rate modulation. We also show that the CTC is sensitive to the jitter in the arrival times of spikes to the E cells, and to the degree of I-cell entrainment. We further find that CTC can occur even if the underlying deterministic system does not oscillate; quasicycle-type rhythms induced by the finite-size noise retain the basic CTC properties. Finally a resonance analysis confirms the relative importance of the I cell pacing for rhythm generation. Analysis of whole network behaviour, including computations of synchrony, phase and shifts in excitatory-inhibitory balance, can be further sped up by orders of magnitude using two coupled stochastic differential equations, one for each population. Our work thus yields a fast tool to numerically and analytically investigate CTC in a noisy context. It shows that CTC can be quite vulnerable to rhythm and input variability, which both decrease phase preference.

Linear noise approximation for oscillations in a stochastic inhibitory network with delay.

Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian "delay" case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm.

Well-posedness of a density model for a population of theta neurons.

Population density models used to describe the evolution of neural populations in a phase space are closely related to the single neuron model that describes the individual trajectories of the neurons of the population and which gives in particular the phase-space where the computations are made. Based on a transformation of the quadratic integrate and fire single neuron model, the so called theta-neuron model is obtained and we shall introduce in this paper a corresponding population density model for it. Existence and uniqueness of a solution will be proved and some numerical simulations are presented.

Synchronization of an excitatory integrate-and-fire neural network.

In this paper, we study the influence of the coupling strength on the synchronization behavior of a population of leaky integrate-and-fire neurons that is self-excitatory with a population density approach. Each neuron of the population is assumed to be stochastically driven by an independent Poisson spike train and the synaptic interaction between neurons is modeled by a potential jump at the reception of an action potential. Neglecting the synaptic delay, we will establish that for a strong enough connectivity between neurons, the solution of the partial differential equation which describes the population density function must blow up in finite time. Furthermore, we will give a mathematical estimate on the average connection per neuron to ensure the occurrence of a burst. Interpreting the blow up of the solution as the presence of a Dirac mass in the firing rate of the population, we will relate the blow up of the solution to the occurrence of the synchronization of neurons. Fully stochastic simulations of a finite size network of leaky integrate-and-fire neurons are performed to illustrate our theoretical results.

Population density models of integrate-and-fire neurons with jumps: well-posedness.

In this paper we study the well-posedness of different models of population of leaky integrate-and-fire neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the different behaviours of the model with jumps compared to its diffusion approximation.