A spiking neural network model for fractional proprioceptive encoding of limb posture and movement in insects

Abstract

Proprioception is key to all behaviours that involve the control of force, posture or movement. Computationally, many proprioceptive afferents share three features: First, their strictly local encoding of stimulus magnitudes causes range fractionation in sensory arrays. As a result, encoding of large joint angle ranges requires convergence of afferent information onto first-order interneurons. Second, their phasic-tonic response properties lead to fractional encoding of the fundamental sensory magnitude and its derivatives (e.g., joint angle and angular velocity). Third, the distribution of disjunct sensory arrays across the body implies that complex movements involve information from multiple joints or limbs. The present study proposes a multi-layer spiking neural network for distributed computation of whole-body posture and movement. The first part of the study models strictly local, phasic-tonic encoding of joint angle by proprioceptive hair field afferents by use of Adaptive Exponential Integrate-and-Fire neurons. Fractionally encoded afferent information about single-joint posture and movement converges on two types of first-order interneurons, tuned to encode either joint angle or velocity across the entire working range with high accuracy. In velocity-encoding interneurons, spike rate increases linearly with angular velocity. The companion paper exploits this distributed position/velocity encoding in second- and third-order interneurons, using combinations of two or three position/velocity inputs from disjunct arrays. The encoding properties of all interneuron layers are evaluated with experimental data on whole-body kinematics of unrestrained stick insect locomotion, comprising concurrent joint angle time courses of [Formula: see text] leg joints. The hierarchical model allows increasingly complex encoding of posture and movement, from angular velocity of a single joint, to movement cycle phases of an entire limb, to parameters of overall body posture.

Keywords: Body posture; Hair field; Insect locomotion; Proprioception; Sensory adaptation; Sensory encoding; Spiking neural network.

Fig. 1

Distributed proprioceptive encoding of leg posture and movement A. The six legs of the Indian stick insect Carausius morosus have similar size and structure. The dataset used here includes joint angles of the thorax-coxa, coxa-trochanter and femur-tibia joints, referred to as the , and joints, respectively. Red arrows indicate locations of proprioceptive hair fields at the and joints of the legs (labeled Cx and Tr, respectively) as well as the head-scape and scape-pedicel joints on the antenna (labeled Sc and Pd, respectively). B. Schematic representation of a proprioceptive hair field on the cuticle near a joint, before and during the joint membrane folds over the hair field during a change in joint angle. A change in joint angle causes hair deflection. As the joint angle increases, the number of hairs deflected and the deflection angle per hair also increase. Each hair acts as a lever arm connected to a dendrite of a sensory neuron. Mechanotransduction channels open, allowing the sensory neuron to spike in proportion to the hair deflection (Tuthill and Wilson 2016). C. Example time course of the joint angle for the joint of the right anterior leg (R1). D. Proposed SNN architecture for distributed proprioception of limb kinematics and body posture. The joint angle time courses , and for each of the six legs are converted into hair angle time courses (not shown) that are sensed by one mechanoreceptor per hair in Layer 1. Solid and open circles refer to antagonistic arrangement of two hair fields per joint. In Layer 2, two velocity (vel +/-) and position (pos +/-) INs per joint encode the posture and movement. In the companion paper, Layer 3 integrates converging information from all joints per limb, and Layer 4 integrates converging information from all limbs

Fig. 2

Bi-directional hair field and extended bi-directional hair field. A. The bi-directional hair field models pairs of hair rows on opposite sides of the joint and working-ranges below (red) and above (blue) the neutral angle . In the idealised hair row, hair angle is a function of joint angle with parameters , , , and (solid red lines), and , , , and (dotted blue lines). The positively and negatively oriented hair angles are calculated using Eq. (1) and Eq. (4), respectively. The resting angle is at . B. Extended bi-directional hair plate: This hypothetical scenario is modified from panel A as follows: , and . Overlap ranges have been omitted for clarity.

Fig. 3

Phasic-tonic spike response of the real and modeled mechanosensory neuron. A. Spike response of mechanosensory afferents from the lateral scapal hair plate in the American cockroach (Okada and Toh 2001). The plot shows the spike rate over time for a tactile hair deflected by a ramp-and-hold function with a constant velocity of from to various end angles. B. Response to ramp-and-hold deflection of a tactile hair with a constant hold angle of , but varying angular velocities. C. Stimulus protocol (top) and corresponding spike responses of the AdEx model (bottom). The stimulus protocol is the same as for A, except with four-fold ramp velocity and shorter hold time. D. Stimulus protocol (top) and corresponding spike responses of the AdEx model (bottom). The stimulus protocol is the same as for B, but with four-fold ramp velocities and shorter hold time. E. Heatmap illustrating the mean absolute error (MAE) between significant spike rates (peak and steady-state) of the modeled and experimental sensory neuron for a parameter sweep across and b. The MAE was computed according to Eq. (12)

Fig. 4

Position IN results. A. Raster plot displaying the response of a hair plate () to a 5-second movement sequence of the joint (thorax-coxa) of the right front leg (R1). Spike events from proprioceptive hairs in the anterior (blue dots) and posterior (red dots) parts of the joint’s working range encode increasingly protracted or retracted postures, respectively. The high spike density results in the dots merging into a continuous line. B. Heat map depicting the mean squared error (MSE) between experimentally obtained joint angles and model predictions across a parameter sweep of and . The results are averaged over 78 trials and 18 joints, totaling 1404 joint angle time courses. C. Spike rate time courses of the and INs in response to the same joint angle movement as shown in panel A. D. Combined and z-normalized spike rates of the and INs from panels A and C closely follow the joint angle time course. The model response and experimental data are plotted in arbitrary units due to z-normalization.

Fig. 5

Velocity IN results. A. For velocity INs, the output spike rate increases linearly with angular velocity. and were varied to test their impact on the spike rate. The four variants shown here differ in threshold velocity for reliable encoding and the slope of the linear relationship. The value of a linear regression fit is given for each variant. B. Spike events superimposed on the joint angle time course reveal consistent encoding of positive slope by INs (blue) and of negative slope by INs (red). C. The spike rate of the and INs (red and blue, respectively) in response to a concurrent change in joint angle velocity (black). D. Combined and z-normalized spike rate of both the and INs in response to a representative experimental angular velocity record. Both the model response and experimental data are plotted in arbitrary units due to z-normalization. A persistent time lag of approximately 0.025 s is observed in the model response.