Fig 1.
Diagram of our model of the dorsal horn (DH) circuit (within the dashed rectangle) including connections between the neuron populations I, E, and P, the afferent fibers Aβ, Aδ, and C, and the dorsal root ganglion (DRG).
We denote inhibitory connections with bars (in red) and excitatory connections with arrows (in blue). The dash-dotted line represents an inhibitory interneuron population (I2) that is modeled indirectly.
Fig 2.
Response curve and NMDA activation plots [see Eq (3)].
A: Response functions of the projection (blue), excitatory (green), and inhibitory (red) neural populations for varying average input firing rates (on the x-axis). Parameters for these curves were chosen to match experiments presented in [21]. B: Activation curve for the NMDA-mediated synaptic input (M∞), shown as a fraction of the maximum synaptic weight gNMDA.
Fig 3.
Simulated response of the populations of afferent nerve fibers to a brief nociceptive stimulus at t = 0.5 s (red arrow).
A: Raster plots of spiking activity of (top) 90 Aβ-, (middle) 90 Aδ-, and (bottom) 820 C-fibers with differing conductance speeds. B: The smoothed instantaneous firing rate (i.e., fAβ(t), fAδ(t), and fC(t)) for each fiber population.
Fig 4.
Daily rhythm in the modulation of pain sensitivity: Experiments and model.
A: Prototypical human “daily pain sensitivity” (i.e., daily changes in pain sensitivity relative to mean pain sensitivity) function (f(x) = 11sin(0.25x + 2.8), where x ∈ [0, 24] hours) fitted to (symbols) data (R2 = 0.73 and RMSE = 4.69) from four experimental studies of pain responses. For more details and sources of these data, see [6]. B: Daily modulation of the stimulus-induced firing rate of the afferent fibers modeled by Eqs (4) and (5). Top(bottom) panel displays the daily modulation of the peak stimulus-induced firing rate of the Aβ- (C-) fibers. In the bottom panel, the blue curve represents the effective modulation of the C-fibers including Aβ-dependent presynaptic inhibition. The x-axis refers to hours since the typical or scheduled morning wake time.
Fig 5.
Daily rhythmicity of pain sensitivity: Comparing model to experiments.
A: Percent of the mean response of the model output (blue) plotted as mean (curve) and standard deviation (shaded region) over 30 realizations of the Poisson input. The fitted curve from Fig 4A is plotted in black open circles. B: Firing rate of the P population in response to the C-fiber input as a function of the time of day. The x-axis refers to hours since the typical or scheduled morning wake time.
Fig 6.
An example output of the PNs at two time points averaged over 30 realizations.
The average firing rate of the P (projection neuron) population during (left) afternoon (lowest pain sensitivity) and (right) early morning (highest pain sensitivity) in response to a brief nociceptive stimulus and modulation in sensitivity of the afferent fibers over the day. The thick curve denotes the mean, the shaded region the standard deviation, of 30 realizations of the Poisson spiking activity on the afferent fibers (for one realization, see Fig 3). We interpret P firing rate frequencies higher than 25 Hz (dashed line) as painful.
Fig 7.
Modeling the wind-up phenomenon.
A: Simulated average firing rate of the P population (top) and NMDA synaptic weight, gNMDA (bottom), in response to a repeated brief nociceptive stimulus (at 2 Hz). In the top panel, the blue curve denotes the mean and the shaded region denotes the standard deviation in response to 20 realizations of the stimulus induced activity of afferent fibers. B: Pain latency computed for a repeated stimulus at 2 Hz. Latency is defined here as the first time when the average firing rate of the P population exceeds the threshold of 25 Hz (interpreted as painful) and decreases as a function of the stimulus index (solid lines indicate times when average firing rate of P exceeds painful threshold). The latency dynamics match those found in experiments [47].
Fig 8.
Characterizing how wind-up phenomenon changes with frequency.
A: The mean of the average firing rate of P neurons during the C-response (which we define as the time interval 90-300 ms after the start of the stimulus, see blue box on bottom of Fig 6) for each stimulus index increases as a function of the stimulus frequency. B: Latency, defined here as the first time when the average firing rate of P population exceeds the threshold of 25 Hz (interpreted as painful), decreases with the change in the frequency of the repeated stimulus.
Fig 9.
Inhibition of painful response by subsequent activation of Aβ-fiber activity.
A: Firing rate of the projection neurons in response to secondary Aβ stimulation following brief nociceptive stimulus activating all three fibers. B: Spike raster plot generated from the model output shown in A for 1000 PNs constructed to resemble Fig 5 in [21], which replicates experiments from [47]. We denote the arrival of the second Aβ-pulse with a red tick mark (asterisk) in A (B).
Fig 10.
Quantifying effectiveness of pain inhibition across the day.
A: Percentage of the baseline firing rate of the PNs as a function of the delay time of the second Aβ pulse at 8 hours following the typical morning wake time. B: Percentage of the painful response as a function of both time of day and delay time of the second Aβ pulse. Color scale indicates the percent of the baseline firing rate of the PNs with no pain inhibition (i.e., 0 s delay), with darker colors representing larger decreases in pain as a result of the pain inhibition.
Fig 11.
A change in the daily rhythms of pain sensitivity in neuropathic pain conditions compared to normal conditions.
A: Daily variation of the stimulus response firing rates of the Aβ- (top panel) and C-fibers (lower panel, dashed curve), and effective C-fiber stimulus response firing rate including effects of Aβ-dependent presynaptic inhibition under normal conditions (lower panel, blue curve, same as in Fig 4B) and Aβ-dependent presynaptic excitation under neuropathic (red curve) conditions. B, C: Daily variation in the response of the PNs to a brief nociceptive stimulus quantified by the percent of its mean (B) and average firing rate of the C-fiber response (C) for normal (blue, same as in Fig 5) and neuropathic (red) conditions. The x-axis refers to hours since the typical or scheduled morning wake time.
Fig 12.
Investigating the weighting parameter, .
A: Percent of the mean and B: average firing rate of the PNs in response to changes in the strength of excitation, during neuropathy, from the Aβ- fibers to the C-fibers, (denoted by gAβC in the figure). The x-axis refers to hours since the typical or scheduled morning wake time.