Performance limitations of relay neurons

Abstract

Relay cells are prevalent throughout sensory systems and receive two types of inputs: driving and modulating. The driving input contains receptive field properties that must be transmitted while the modulating input alters the specifics of transmission. For example, the visual thalamus contains relay neurons that receive driving inputs from the retina that encode a visual image, and modulating inputs from reticular activating system and layer 6 of visual cortex that control what aspects of the image will be relayed back to visual cortex for perception. What gets relayed depends on several factors such as attentional demands and a subject's goals. In this paper, we analyze a biophysical based model of a relay cell and use systems theoretic tools to construct analytic bounds on how well the cell transmits a driving input as a function of the neuron's electrophysiological properties, the modulating input, and the driving signal parameters. We assume that the modulating input belongs to a class of sinusoidal signals and that the driving input is an irregular train of pulses with inter-pulse intervals obeying an exponential distribution. Our analysis applies to any [Formula: see text] order model as long as the neuron does not spike without a driving input pulse and exhibits a refractory period. Our bounds on relay reliability contain performance obtained through simulation of a second and third order model, and suggest, for instance, that if the frequency of the modulating input increases or the DC offset decreases, then relay increases. Our analysis also shows, for the first time, how the biophysical properties of the neuron (e.g. ion channel dynamics) define the oscillatory patterns needed in the modulating input for appropriately timed relay of sensory information. In our discussion, we describe how our bounds predict experimentally observed neural activity in the basal ganglia in (i) health, (ii) in Parkinson's disease (PD), and (iii) in PD during therapeutic deep brain stimulation. Our bounds also predict different rhythms that emerge in the lateral geniculate nucleus in the thalamus during different attentional states.

Figure 1. A relay neuron.

(A) Illustrating a relay neuron. Ensemble activity of all the distal synapses (stars) is modulating input . The proximal synapses (diamonds) form the driving input . The output is the axonal voltage . (B) A block diagram of a relay neuron showing two inputs and output .

Figure 2. Properties properties of .

(A) Illustrates the equilibrium point , the steady state orbit and the orbit tube, , for given by (3) and . The orbit tube is shown for . (B) Illustrates , the threshold voltage and threshold current . Note that these parameters are defined by the undriven system (9). (C) Illustrates the critical hypersurface , a successful response trajectory, an unsuccessful response trajectory, and the refractory zone, for the undriven system (9). The time it takes for the solution to leave after generating a successful response is called the refractory period, . Note that refractory zone depends on and therefore also depends on . Additionally, note that the region shaded in the darker grey is also in the refractory zone, because if is in this region then such that Therefore, a successful response cannot be generated if is in this region by definition. (D) Dependence of on . Note that is approximately a straight line with slope , i.e . (E) Illustrates vs and .

Figure 3. Threshold.

Illustrates the critical hypersurface , which defines the threshold for a successful response.(9) generates a successful response for any initial condition that is to the right of the hypersurface i.e. . Whereas, any initial condition to the left of the hypersurface results in unsuccessful response.

Figure 4. Calculation of .

Illustrates and . When an pulse arrives, the solution jumps from to . Now, whether the neuron generates a successful response or not is governed by the local dynamics. Therefore, we linearize (4) about to analyze the behaviour of for . If a successful response is generated, such that else if an unsuccessful response is generated such that .

Figure 5. R vs .

Plots the theoretical and numerically computed reliability as a function of , with . The dotted lines are the lower and upper bounds on reliability from the (48) and (47), respectively. The solid line is calculated by running simulations of (1), and the error bars indicate .

Figure 6. vs and - A.

Plots the theoretical and numerically computed reliability as a function of , with . B. Plots the theoretical and numerically computed reliability as a function of with , . The dotted lines are the lower and upper bounds on reliability from the (48) and (47), respectively. The solid line is calculated by running simulations of (4), and the error bars indicate .

Figure 7. Dependence of and on model parameters.

A. Plots as a function of B. (see (35) versus and . Note that depends largely upon , whereas its dependence upon is minimal. changes the maximum value of but does not effect it much in the high frequency range.

Figure 8. A. Voltage profile of the 3rd order model in the bursting mode ( ) B. zoomed in view of a burst C. vs for the order model.

In this Figure, we illustrate the results from a 3rd order model of a thalamic neuron. A. Plots the voltage profile obtained from the model in response to pulses in . Note that each pulse in either generates a burst of spikes or does not spike at all. B. Zoomed in view of a burst. C. Plots the theoretical and numerically computed reliability as a function of , with ,,. The dotted lines are the lower and upper bounds on reliability from the (48) and (47), respectively. The solid line is plots calculated by running simulations of (4), and the error bars indicate . We estimated as the minimum height of a pulse that makes the neuron generate a successful response.

Figure 9. A. Voltage profile of the 3rd order model in the tonic mode ( ) B. zoomed in view of a spike C. vs for the order model.

In this Figure we illustrate the results from a 3rd order model of a thalamic neuron. A. Plots the voltage profile obtained from the model in response to pulses in . Note that each pulse in either generates a successful spike or generates unsuccessful spike. B. Zoomed in view of a successful spike. C. Plots theoretical and numerically computed reliability versus , with , , , , , . The dotted line is plotting the lower and upper bounds on reliability from the (48) and (47), respectively. Note that here , therefore . The solid line plots calculated by running simulations of (4), and the error bars indicate . We estimated as the minimum height of a pulse that makes the neuron spike.

Figure 10. Thalamocortical loop in motor signal processing.

(A) Simplified view of basal ganglia thalamo-cortical motor signal processing. Sensorimotor cortex generates the driving input and projects to the motor thalamus. The thalamus relay of cortical input is modulated by the basal ganglia (BG). (B) Relay reliability curves computed from our analysis as a function of and from (49). (C) Simulations of (basal ganglia output) from the computational study for the Healthy, PD and PD with high frequency deep brain stimulation (HFDBS) cases. As we can see in the healthy case, the amplitude of the BG output, , is smaller compared to the PD BG output, resulting in a higher relay reliability. HFDBS increases the frequency, , of the BG output, resulting in a higher relay reliability. (D) Intuition of how reliability changes in the three cases. In PD, is larger, therefore, the diameter of the orbit tube is larger compared to the orbit tube for healthy. This results in more time spent in the unsuccessful response region , which leads to poor reliability. In contrast, in PD case with HFDBS applied, is larger and the gains decrease, which generates a smaller orbit tube. In this case, the state spends more time in the successful response region of the orbit tube, resulting in high reliability.