Fig 1.
Experimental data from Jepma et al., [20] (left) and our Kalman filter simulations results (right). Placebo hypoalgesia (or the reverse effect, nocebo hyperalgesia) is the phenomenon where identical noxious stimuli may result in lower (higher) perceived pain (, filled circles) depending on expectations (
, open triangles). In classical conditioning the expectations are influenced by cues that have been associated with high or low noxious stimuli (‘high cue’ and ‘low cue’). In testing the effect of conditioning Jepma et al., applied two different levels of noxious heat stimuli (‘low heat’ and ‘high heat’). Note that the during the test trials level of noxious stimuli is independent from the cue, i.e., the low cue is paired with both high and low thermal stimulation. a) and b) average expected (open triangles) and perceived (filled circles) pain as a function of cue type and trial for experimental and simulated data, respectively. c) and e) perceived (filled circles,), d) and f) expected (open triangles) pain as a function of stimulus temperature and cue type for experimental and simulated data, respectively. Error bars indicate inter-individual standard errors. Note that the experimental data is measured on a 100-unit scale, whereas the simulated data is on an 11-unit scale.
Fig 2.
Results of the Kalman filter simulations of chronic pain.
The model produces output reflecting chronic pain when there is elevated uncertainty in the sensory input, in combination with the internal model parameter (solid blue line in upper plots). Left: when R (dash-dotted, purple line in upper plot) is low the perceived pain,
, (solid red line in lower plot, the shaded area indicates the interquartile range) will primarily be influenced by the sensory input and closely correspond to the true level of tissue damage, x, (dash-dotted black line in lower plot).Right: for a larger value of R, predictions from the internal model have a stronger influence on the perceived level of pain, possibly resulting in a chronically elevated level of pain even after the tissue damage has recovered.
Fig 3.
Results from Kalman filter simulations of neuropathic pain.
Damage to the sensory nervous system may result in increased uncertainty of sensory input relating to the level of tissue damage. Depending on the level of uncertainty (value of R, dash-dotted purple line in upper plots) and the value of the internal model parameter (solid blue lines in upper plots), these changes could result in spontaneous neuropathic pain (solid red lines in lower plot, the shaded area indicates the interquartile range).The initial overshoot for
in the two rightmost plots, where R = 802 and R = 8002, is caused by the initial variance of the estimate P, being low. When P≪R, the posterior estimate is dominated by the prior, which here predicts increasing pain since
. Due to the uncertainty in the sensory input, the value of P rapidly increases, until an equilibrium is reached, in which the posterior still is strongly influenced by the prior, but also has some influence from the noisy sensory input.
Fig 4.
Results of the hierarchical Kalman filter simulations of classical conditioning.
We simulate the conditioning procedure similar to the learning phase of the experimental paradigm described by Jepma et al., [20]. During conditioning previously neutral cues (‘high cue’ and ‘low cue’ in the figures) are repeatedly paired with high or low noxious stimuli, resulting in diverging values of internal model parameters and
(panel a)), and creating expectations of high pain associated with the high cue, and low pain associated with the low cue (panel b)). In testing the effect of conditioning the cues are paired with intermediate intensity noxious stimuli (47°C for ‘low heat’ and 48°C for ‘high heat’). Note that the during the test trials the level of noxious stimuli is independent from the cue, i.e., the low cue is paired with both high and low thermal stimulation. a) median value of
(dashed red line) and
(dashed blue line) across conditioning trials. Shaded areas indicate the interquartile range. b) average expected pain for high-cue trials (red) and low-cue trials (blue) during conditioning. Open triangles indicate the expected pain for each participant on each conditioning trial. c) the average expected (
, open triangles) and perceived (
, filled circles) pain as a function of cue type on each test trial. d) perceived (filled circles) and e) expected (open triangles) pain as a function of stimulus temperature and cue type. Error bars indicate inter-individual standard errors.
Fig 5.
Results from hierarchical Kalman filter simulations of how the value of (solid blue lines in upper plots) at the time of a nerve injury and the level of sensory disruption (i.e., value of R, dash-dotted purple line in upper plot) may play a role in the characteristics of subsequent neuropathic pain.
Simulations were run for three different initial values of . The top panels show the inferred value of
over time, and the bottom panels show the corresponding level of perceived pain,
. For R = 0.82 (left), the perceived pain (solid red lines in the lower plots) is strongly influenced by sensory input and remains close to 0. For R = 82 (second from the left),
regardless of the initial value, and the perceived pain is tonically at an intermediate intensity with little variance. For R = 8002 (right), the value of
is no longer at all influenced by sensory input and reduces to a random walk centered at the initial value
and variance equal to the noise in the internal model. In this scenario, if
the pain is likely to stay tonically high, and similarly if
the level of pain is likely to drop to stay at 0. If
, the perceived level of pain may vary widely as the value of
fluctuates above and below 0. For R = 802 (second from the right) the model displays a mixture of the behavior described for R = 82 and R = 8002. The shaded areas indicate the interquartile ranges.
Fig 6.
Results from hierarchical Kalman filter simulations of how the value of the internal model parameters are determined by previous painful experiences and may contribute to neuropathic pain following a nerve injury.
In the left panels, A = 0.99, yielding persistent pain (solid red line in lower plots) following noxious stimuli (indicated by stars). This value of A results in (solid blue line in the upper plot), and a high risk of spontaneous neuropathic pain following a nerve injury (corresponding to a change in the value of R, indicated by the vertical dashed line). In the middle panels A = 0.9, resulting in quickly transient pain, a lower value of
and a lower risk of spontaneous neuropathic pain. In the right panels A = 0.9 again, but in this simulation, there are no noxious stimuli. These circumstances result in
and a higher risk of spontaneous neuropathic pain than in the example in the middle. The shaded areas indicate the interquartile ranges.
Fig 7.
Results of the hierarchical Kalman filter simulations of offset analgesia.
Offset analgesia is defined as a disproportionally large pain decrease after a minor noxious stimulus intensity reduction. This phenomenon is often elicited by applying noxious thermal stimulation of the same temperature in time intervals T1 and T3, separated by time interval T2 with slightly higher temperature. Our simulation results show a qualitatively similar underdamped response in perceived lever of pain (open red circles) to the minor temperature reduction as is observed in experimental studies of offset analgesia. In the control condition a constant temperature is applied throughout all three time intervals, resulting in a constant level of pain (filled red circles). Error bars indicate the inter quartile ranges.
Fig 8.
Schematic figure of the Kalman filter model, depicting the evolution of states across iterations and the relationship between model variables and parameters.
The related variance/noise/covariance to each parameter is indicated next to the arrows. “Top-down” predictions, , are formed by the control input, u, and the internal model,
, where
,. The prediction is combined with “bottom-up” sensory input, z, to form the final estimate,
, which, in our model, reflects the perceived level of pain.
Table 1.
Overview of the variables and parameters in the single-layer Kalman filter along with dimensionality and, where applicable, the default value used in simulations when nothing else is specified.
Fig 9.
Schematic figure of the hierarchical Kalman filter model, depicting the evolution of states and internal model parameters across iterations, and the relationship between model variables and parameters.
The related variance/noise/covariance to each parameter is indicated next to the arrows. State estimation (lower part of the figure with arrows in solid black lines) is comparable to the single-layer Kalman filter model depicted in Fig 8. The estimation error, i.e., the difference between the predicted and perceived pain, , acts as the “bottom-up” input to the parameter estimation Kalman filter (upper part of the figure with arrows in dashed grey lines).The estimated parameters from the previous iteration (and additive internal model noise, Qp) act as the prior estimate,
. Finally, the prior estimate,
, and the estimation error,
, are combined to form the new estimated internal model parameters.
Table 2.
Overview of the variables and parameters in the hierarchical Kalman filter along with dimensionality and, where applicable, the default value used in simulations when nothing else is specified.
Note that I(1+m) denotes a (1+m)-dimensional identity matrix.
Fig 10.
Pain is assessed in the intervals {T1, T2, T31, T32, T33, T34} in the offset analgesia simulations.
The first period following ramp-up or ramp-down of temperature, indicated by Δ in the figure, is omitted to account for any delay in change of perceived pain relative to the change in temperature.