Table 1.

Literature on mathematical models of pain

Publication	Pain Type	Model Types* [13]	Summary
Minamitani and Hagita (1981) [14]	Nociceptive	Analogous/ Symbolic	The neural network model simulated the conduction mechanism of pain and touch sensations. Although only one directional ascending and descending pathway for pain sensation was represented, and no interaction from inhibition or facilitation was considered, the modalities of graded touch sensation and two different pain modalities were observed.
Britton and Skevington (1989) [15]	Nociceptive	Analogous/ Symbolic	Melzack's gate control theory of pain was translated into a mathematical model simulating acute pain for a single transmission unit. The partial differential equations were based on the Wilson-Cowan model for synaptically coupled neuronal networks.
Spitzer et al. (1995) [27]	Neuropathic (Phantom Limb)	Analogous/ Phenomenological	A self-organization feature map using Kohonen network was used to simulate the effects of amputation. The Kohonen network was trained on input patterns and subsequently deprived parts of the input patterns in order to simulate partial deafferentation. This led to reorganization driven by input noise, which represented noise generated by erratic firing of lacerated dorsal root ganglion sensory neurons.
Haeri et al. (2003) [28]	Nociceptive	Analogous/ Phenomenological	An artificial neural network to model the steady state behavior of pain mechanisms was developed using input patterns from small and large nerve fibers. For stimulation states corresponding to acute pain, a collection of basic patterns was used as features for the model. Given a novel pain stimulus, the prediction of pain was possible.
Xu et al. (2008) [29]	Nociceptive (Thermal)	Symbolic	Considering the biophysical and neural mechanisms of pain sensation, a mathematical model for quantifying skin thermal pain that included transduction, transmission, and perception was proposed. This model proposed that the intensity of thermal pain was related to the character of the noxious stimulus.
Cecchi et al. (2012) [30]	Nociceptive (Thermal)	Analogous/ Symbolic	Thermal pain perception was modelled as a dynamical system to be compared to reported pain ratings from intensity-varying thermal stimuli. Using a sparse regression method, pain ratings were predicted according to fMRI data and reported pain ratings.
Rho and Prescott (2012) [31]	Neuropathic	Conceptual/ Symbolic	A computational model was developed to simulate the onset of neuronal hyperexcitability from a normal spiking pattern. Parameters changes were sufficient to alter the normal spiking pattern to a repetitive one, enabling membrane potential oscillations, and bursting, suggesting that the three pathologies are related.
Boström et al. (2014) [32]	Neuropathic (Phantom Limb)	Conceptual/ Phenomenological	A computational model of phantom limb pain was developed based on the increase of spontaneous nociceptive firing. They proposed that the same underlying mechanism that results in ectopic spontaneous activity of deafferented nociceptive channels was responsible for phantom pain, maladaptive reorganization, and persistent representation.
Prince et al. (2014) [33]	Nociceptive	Analogous/ Symbolic	Britton and Skevington's acute pain model was replicated and expanded to verify the assumption that neighboring transmission units behave similarly. With sufficient increase in the number of transmission units input to the midbrain, transmission unit potential decreased, suggesting a saturation point in which transmission units may fail to fire despite neural fiber activation.
Tigerholm et al. (2014) [34]	Nociceptive	Analogous/ Symbolic	Axonal conduction velocity by activity differs between patients with neuropathic pain and those without, suggesting that this property may play a role in the development of neuropathies. A mathematical model of human cutaneous C-fibers was developed to investigate the activity-dependent changes of axonal spike conduction.
Dick et al. (2017) [35]	Nociceptive	Analogous/ Symbolic	By implementing a mathematical model of rat nociceptive neuronal membrane, a mechanism of ectopic bursting suppression in dorsal root ganglia neurons with comenic acid was proposed. The administration of comenic acid to the model reduced rhythmic discharge frequency due to a decrease in the effective charge transferring via sodium gate activation dynamics.
Crodelle et al. (2019) [36]	Nociceptive	Statistical/ Symbolic	A mathematical model of the dorsal horn neural circuit relying on firing rates and model parameters from experimental literature was developed to describe daily modulation of pain sensitivity. The inversion of daily rhythmicity of pain in neuropathic patients was proposed to be the result of dorsal horn circuitry dysregulation.
Dick (2020) [37]	Nociceptive	Analogous/ Symbolic	Bifurcation analysis was used to determine the relationship between the nociceptive neuron model and the antinociceptive effect that occurs during neuropathic pain suppression. The molecular mechanism of the bursting suppression was associated with the modification of the activation gating system of Nav1.8 channels by comenic acid, suggesting a possible molecular treatment for neuropathic pain.

^*Conceptual models are the most basic of the model types. They are pedagogical and useful as foundations to more quantitative models. Analogous models borrow their structure from more well-known systems. Symbolic models are ordinarily described in mathematical language, i.e., symbols. Phenomenological models are symbolic in nature, but are often referred to as “black box models,” since their predictive power is the only priority. Statistical models are symbolic models where the mathematics is taken from probability theory. For additional details, see [13].