Analysis of reflex modulation with a biologically realistic neural network

Abstract

In this study, a neuromusculoskeletal model was built to give insight into the mechanisms behind the modulation of reflexive feedback strength as experimentally identified in the human shoulder joint. The model is an integration of a biologically realistic neural network consisting of motoneurons and interneurons, modeling 12 populations of spinal neurons, and a one degree-of-freedom musculoskeletal model, including proprioceptors. The model could mimic the findings of human postural experiments, using presynaptic inhibition of the Ia afferents to modulate the feedback gains. In a pathological case, disabling one specific neural connection between the inhibitory interneurons and the motoneurons could mimic the experimental findings in complex regional pain syndrome patients. It is concluded that the model is a valuable tool to gain insight into the spinal contributions to human motor control. Applications lay in the fields of human motor control and neurological disorders, where hypotheses on motor dysfunction can be tested, like spasticity, clonus, and tremor.

Fig. 1

Scheme of the NMS model. d(t): external force disturbance which act to the limb; x(t): endpoint position; iss(t): supraspinal excitatory commands; F_m(t): muscle force from the contraction dynamics (CD); x_m(t): muscle stretch; : muscle stretch velocity; u(t): motoneuron activation. The muscle spindles (MS) and Golgi tendon organs (GTO) sense the muscle stretch, stretch velocity and force resulting in afferent signals on the I_a(t), I_b(t), and II(t) afferents respectively. These afferent signals have a conduction time τ_I or τ_II, depending on the afferent path, before reaching the BNN. After the conduction time τ_u(t), u(t) reaches the muscle and is converted into the muscle activation a(t) by the activation dynamics (AD). An asterisk (*) denotes the necessity to double the components for the flexor and the extensor muscles

Fig. 2

Geometrical representation of the one-DOF model of the human shoulder. The limb with concentrated mass m_l is actuated by two antagonistic muscles, (1) and (2), and disturbed by a force d(t) resulting in the position deviation x(t). The limb length is represented by l_l and the constant moment arm of the muscle force by l_a. The black figure is the limb in equilibrium or reference position; the grey figure is the limb after a small deviation

Fig. 3

Neuron populations and their interconnections in the BNN of this study (based on Bashor 1998). Proprioceptive input comes through the Ia, Ib and II afferents and the firing rate output of the motoneurons drives the muscles in the musculoskeletal model. The supraspinal commands (i_ss, paths not displayed) excite the BNN via synapses e6, e11, e16 and e24. ESTC are the normal excitatory short time constant synapses, DESTC and TESTC the ESTC with double and triple connection strengths, ELTC the excitatory long time constant synapses and ISTC the inhibitory short time constant synapses. The neuron types are given inside the circles, where MN denotes the motoneurons, RC the Renshaw cells and IA, IB, IN and EX respectively the IA, IB, inhibitory and excitatory interneurons. The synapse identifying numbers are denoted by e.. (excitatory) or i.. (inhibitory) and the value between the brackets represents the number of terminals received from the indicated source by a representative cell

Fig. 4

Block scheme of human postural control expressed in the frequency domain. H_nms(f): transfer function of the dynamics of the NMS model; D(f): external force disturbance; X(f): skeletal bone endpoint position; N(f): model remnant. The NMS dynamics (dashed box) are described by the linear transfer function H_nms(f), together with the remnant N(f), which is uncorrelated with D(f)

Fig. 5

Block scheme of the linear arm model of which the transfer function H_nms(s) is derived. D(s): external force disturbance; X(s): skeletal bone endpoint position; H_int(s): intrinsic (muscle) dynamics; H_ref (s): reflexive dynamics; H_act(s): muscle activation dynamics

Fig. 7

Typical muscle stretch, stretch velocity and muscle force signals (left plots) and the resulting II, Ia and Ib afferent activity (right plots). Dashed line indicates the average signal. The signals were recorded on the flexor side of the BNN during a single simulation with the default BNN setting. To improve readability, only a single second of the simulation is plotted

Fig. 6

Typical neuron activity for the six agonistic neuron populations. Left y-axis: number of neurons firing a spike; right y-axis: related spiking frequency, averaged over the population. The grey horizontal lines indicate the number of neurons firing a spike and the spiking frequency, averaged over the entire simulation run. The signals were recorded on the flexor side of the BNN during a single simulation with the default BNN setting. To improve readability, only a single second of the simulation is plotted

Fig. 8

The 15 most sensitive synaptic and sensory parameters. Top: sensitivity measure S together with the parameter description; bottom: VAF of the lumped parameter model (mean ± SD over the seven variations factors)

Fig. 9

The 15most sensitive neuronal parameters (top) andVAF of the lumped parameter model (bottom)

Fig. 10

Force disturbance (above) and endpoint position displacement (below) of three different simulation runs. Monosynaptic stretch reflex synaptic strengths of 0 (dotted), 1.5 (solid) and 3 (dashed) times the default synaptic strength, simulating decreasing presynaptic inhibition on this synapse. To improve readability, only 4 s of a simulation are plotted

Fig. 11

Gain (above), phase (middle) of the joint admittance, and the coherence (below). Monosynaptic stretch reflex synaptic strengths of 0 (dots), 1.5 (pluses) and 3 (‘x’-marks) times the default synaptic strength, simulating decreasing presynaptic inhibition on this synapse. Solid line: simulation; dotted line: linear model fit; and dashed line: simulation without afferent feedback

Fig. 12

Feedback gain values for the position (k_p), velocity (k_v) and acceleration (k_a), together with the VAF scores, plotted against the monosynaptic stretch reflex synapse strength (relative to default strength). By increasing the monosynaptic stretch reflex synapse strength, a decreasing presynaptic inhibition is simulated; for maximum presynaptic inhibition, the monosynaptic stretch reflex synapse strength is 0. Black lines: normal situation; grey lines: pathological situation, synapse i3 was disabled