From the motor cortex to the movement and back again.

The motor cortex controls motor behaviors by generating movement-specific signals and transmitting them through spinal cord circuits and motoneurons to the muscles. Precise and well-coordinated muscle activation patterns are necessary for accurate movement execution. Therefore, the activity of cortical neurons should correlate with movement parameters. To investigate the specifics of such correlations among activities of the motor cortex, spinal cord network and muscles, we developed a model for neural control of goal-directed reaching movements that simulates the entire pathway from the motor cortex through spinal cord circuits to the muscles controlling arm movements. In this model, the arm consists of two joints (shoulder and elbow), whose movements are actuated by six muscles (4 single-joint and 2 double-joint flexors and extensors). The muscles provide afferent feedback to the spinal cord circuits. Cortical neurons are defined as cortical "controllers" that solve an inverse problem based on a proposed straight-line trajectory to a target position and a predefined bell-shaped velocity profile. Thus, the controller generates a motor program that produces a task-specific activation of low-level spinal circuits that in turn induce the muscle activation realizing the intended reaching movement. Using the model, we describe the mechanisms of correlation between cortical and motoneuronal activities and movement direction and other movement parameters. We show that the directional modulation of neuronal activity in the motor cortex and the spinal cord may result from direction-specific dynamics of muscle lengths. Our model suggests that directional modulation first emerges at the level of muscle forces, augments at the motoneuron level, and further increases at the level of the motor cortex due to the dependence of frictional forces in the joints, contractility of the muscles and afferent feedback on muscle lengths and/or velocities.

Fractional-order leaky integrate-and-fire model with long-term memory and power law dynamics.

Pyramidal neurons produce different spiking patterns to process information, communicate with each other and transform information. These spiking patterns have complex and multiple time scale dynamics that have been described with the fractional-order leaky integrate-and-Fire (FLIF) model. Models with fractional (non-integer) order differentiation that generalize power law dynamics can be used to describe complex temporal voltage dynamics. The main characteristic of FLIF model is that it depends on all past values of the voltage that causes long-term memory. The model produces spikes with high interspike interval variability and displays several spiking properties such as upward spike-frequency adaptation and long spike latency in response to a constant stimulus. We show that the subthreshold voltage and the firing rate of the fractional-order model make transitions from exponential to power law dynamics when the fractional order α decreases from 1 to smaller values. The firing rate displays different types of spike timing adaptation caused by changes on initial values. We also show that the voltage-memory trace and fractional coefficient are the causes of these different types of spiking properties. The voltage-memory trace that represents the long-term memory has a feedback regulatory mechanism and affects spiking activity. The results suggest that fractional-order models might be appropriate for understanding multiple time scale neuronal dynamics. Overall, a neuron with fractional dynamics displays history dependent activities that might be very useful and powerful for effective information processing.