Repetitive antegrade contractions

Fig 3 presents a normal RACs pattern obtained through simulations of the mathematical model, qualitatively reproducing the key elements observed in RACs: repetitiveness, forward (antegrade) propagation, and non-overlapping contractions. The contractions emerge and are sustained autonomously and independently of any externally prescribed input. We establish Fig 3 as the baseline case. In this section, we use the model to explain the triggering mechanism of the pattern and the development of its essential elements, revealing the underlying dynamics of RACs.

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Fig 3. Esophageal spatio-temporal topography.

Depicts non-dimensional color-coded cross-sectional area with respect to A_o, obtained through a neuromechanical model simulation of FLIP manometry. is non-dimensional length with respect to L and is non-dimensional time with respect to . The model exhibits a repetitive, antegrade contraction pattern autonomously triggered, mirroring observations from clinical FLIP measurements. The parametric values used to obtain these results are listed in Table 1.

https://doi.org/10.1371/journal.pcbi.1013778.g003

Repetitiveness.

Excitatory signals from mechanoreceptors (S_E) serve as the primary initiators of RACs and play a pivotal role in sustaining the repetitive pattern. Fig 4A demonstrates the consequence of disabling the mechanoreceptors, resulting in a lack of contractility due to insufficient excitatory input. Fig 4B illustrates a scenario in which the sensation of the mechanoreceptors is disabled, and a brief excitatory input is introduced at the rostral end of the smooth muscle section of the esophagus. In the model’s context, this is equivalent to setting everywhere except the rostral end, where for a short time period. As depicted in Fig 4B, the rostral excitatory input travels down the esophageal length, leading to a single, propagating contraction. However, without the mechanoreceptors, there is no sustained excitatory input to initiate additional contractions.

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Fig 4. Color-coded spatiotemporal maps showing non-dimensional cross-sectional area (CSA), excitatory neuronal activity level, or muscle contraction pattern (), obtained from a neuromechanical model simulating esophageal distension tests.

In the panels, and denote non-dimensional length and time, respectively, normalized by total esophageal length, L, and excitatory time constant, . (A) CSA: Mechanoreceptors are disabled; the esophagus remains quiescent. Achieved by increasing the sensitivity threshold, , or reducing sensory feedback to excitatory neurons, w_E. (B) CSA: Mechanoreceptors are disabled, and a brief excitatory input (equivalent to w_E) is applied at the rostral end, generating a single contraction before recovery. (C) Excitatory neuronal population activity: Eliminating neuronal neighboring connections by setting b = d = 0, resulting in uniform excitation along the esophagus. (D) CSA: Reducing inhibitory signaling from the anterior inhibitory population, d, increases excitation duration and produces overlapping contractions. (E) CSA: Lowering the excitatory activation threshold, , generates early excitatory spikes and overlapping contractions. (F) Muscle contraction pattern : Simulated sustained esophageal distension using a short bag, but distension is set to activate inhibitory neurons locally only, such that S_I = 0 caudal to the bag. Parameter values for each simulation are listed in Tables A-E in S3 Text.

https://doi.org/10.1371/journal.pcbi.1013778.g004

The emergence of the repetitive, rhythmic attribute observed in FLIP studies is mathematically elucidated through the model. Consider section i, positioned along the length of the distended esophagus. Adequate sustained volumetric distension around i triggers the activation of local distension-sensitive mechanoreceptors. These receptors, in turn, excite excitatory neurons at i (via S_E,i), initiating the excitatory phase. The activated excitatory neurons then stimulate both excitatory and inhibitory cells at location i. Over time, the activity of inhibitory cells surpasses a threshold level, overcoming both excitatory and inhibitory activity and inducing a refractory period where all cells cease activity. The sustained distension results in constant excitation, initiating a new cycle when all cells are inactive [51]. Note that distension-induced excitation also occurs in inhibitory pathways (S_I), ensuring that inhibition precedes excitation.

Comparison with clinical data

Numerous studies over the years explored esophageal responses to distension, offering insights into the typical reactions of the esophagus under various conditions. Similar to RACs, many of these responses are non-intuitive and their mechanisms have not been formally explained [14,16,60–63]. In this section, we showcase the capability of the proposed neural model to reproduce these diverse distension-induced esophageal scenarios, demonstrating its versatility beyond FLIP manometry. Importantly, we leverage the results to briefly explain these non-intuitive clinical observations in a healthy esophagus. We present four distinct scenarios representing a healthy esophageal response to distension:

1. Case 1. Transient esophageal distension: Results in a single contraction [62]. The short stimulus can be introduced through abrupt inflation and deflation of a distending medium or through vagal efferent nerve stimulation.
2. Case 2. Prolonged esophageal distension using a short balloon followed by abrupt deflation: During the distension, the esophagus caudal to the distended section is mostly inactive. Upon abrupt deflation, one phasic contraction appears, traveling down the length of the esophagus [62]. The pressure and cross-sectional area variations along the distended region are not reported in these studies.
3. Case 3. Sustained distension of a short section of the esophagus: Results in repetitive contractions along the distended section, as observed through a pressure probe located inside the fluid-filled distending bag [16].
   The contractions’ frequency remains independent of the bag’s length, exhibiting a consistent rate of 6 contractions per minute, akin to the FLIP scenario [16,41]. The pressure variations caudal to the distended region are not reported in these studies.
4. Case 4. Sustained esophageal distension along its entire smooth muscle length: Results in RACs [41,44].

When reproducing the above scenarios, the neural circuit and its parametric values remain constant across the different simulations. The objective is to demonstrate that the properties inherent in the baseline solution, representing a typical esophageal response to FLIP (Fig 3), automatically encompass all clinically observed distension-induced scenarios. The simulations vary by adjusting the distension’s location and duration along the esophagus.

All four scenarios are successfully reproduced. Cases 1 and 4 are displayed in Figs 4B and 3, respectively, and discussed in the preceding section. Fig 4B corresponds to Fig 1 in the manuscript by Paterson et al. [62]. Cases 2 and 3 are presented in Fig 5 and elaborated upon in the following discussion. Fig 5A corresponds to Fig 2 in the manuscript by Paterson et al. [62], and Fig 5B corresponds to Fig 1D in the manuscript by Gregersen et al. [16].

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Fig 5. Color-coded spatio-temporal topographies depicting muscle contraction patterns (, non-dimensional by definition) obtained by the neuromechanical model, simulating two distinct esophageal distension tests.

is non-dimensional length with respect to L and is non-dimensional time with respect to . (A) Prolonged balloon distension followed by abrupt deflation (Case 2). (B) Sustained esophageal distension using a short bag (Case 3). The parametric values used for these simulations are displayed in Tables F and G in S4 Text.

https://doi.org/10.1371/journal.pcbi.1013778.g005

Note that cases 2 and 3, though similar, were reported by different research groups, conducting different studies. Paterson et al. [62] (Case 2) focused on the esophageal response caudal to the distension (air-filled balloon), providing detailed pressure readings solely caudal to the balloon. Gregersen et al. [16] (Case 3) documented the pressure and cross-sectional area only inside the distending medium. From these separate studies, we infer that sustained esophageal distension using a short bag or balloon induces repetitive contractions only along the distended section, with the rest of the esophagus mostly inactive. Upon abrupt removal of the distension, a single contraction emerges, traveling beyond the distended segment. Fig 5 displays the muscle contraction pattern over the length of the smooth muscle section of the esophagus, capturing the behavior inside and outside the bag/balloon.

To explain the clinical observations in cases 2 and 3, we specifically focus on unique aspects not discussed in the preceding section, including (i) the selective appearance of contractions solely along the distended section (with a quiescent state caudal to the bag), (ii) the observation of a traveling contraction upon deflation of the bag, and (iii) the explanation for the contraction frequency’s independence of the bag’s length.

1. (i) As previously established, repetitive muscle contractions at any location along the smooth muscle region of the esophagus are both initiated and sustained by local distension-sensitive mechanoreceptors that excite nearby excitatory neurons. Because this input is primarily local [31], regions not directly distended, such as those proximal or distal to the bag, receive little to no excitatory drive and thus remain quiescent (as illustrated in Fig 4A).
   While this explains the lack of contraction proximal to the distended segment, excitatory neurons can also receive input from other excitatory neurons located rostrally (parameter b in our model). This coupling allows excitation to propagate even in the absence of local distension, as shown in Fig 4B. Why, then, do these waves not continue propagating caudally beyond the distended region?
   The key factor is descending inhibition. When the esophagus is excited at any location, excitation is sent to inhibitory neurons caudally [64], causing the esophagus to relax in anticipation of an incoming bolus [10]. Thus, introducing enough inhibition stops the wave from propagating beyond the distended region. In our model, this is represented by , where i is a location caudal to the distended section. If we set the distension-induced excitation to inhibitory neurons only around the distended region (), as in Fig 4F, contractions caudal to the distended region are present.
2. (ii) Note that the inhibitory activity levels caudal to distension are low, allowing a quick recovery once distension is eliminated. Conversely, excitatory activity levels along the distended section are high, taking longer to recover. Therefore, upon abrupt emptying of the bag or balloon, the excitatory signal is free to travel down through neighboring communication (parameter b), unimpeded by inhibitory activity (as depicted in Fig 4B). This propagating excitatory signal translates into muscle contraction, resulting in the observed phasic contraction.
3. (iii) Examining the rate of propagating contractions is equivalent to studying the frequency of muscle contraction at a single location, dependent on the excitation-refractory cycle of neuronal populations at that specific location [16,44]. The time elapsed from the initiation of the excitatory phase to the end of the refractory period is dictated by the neural architecture and the strength of excitatory input from the distension-sensitive mechanoreceptors (w_E) [51]. Since these parameters are independent of the bag’s length, the contraction frequency is not expected to change with the bag’s length.

Esophageal motility disorders

In this section, we showcase how alterations in specific parameters disrupt the RACs pattern. These disruptions result in solutions reminiscent of established esophageal motility disorders, as defined by the Chicago Classification [35,44]. This demonstration not only exposes the underlying mechanisms of these disorders but also emphasizes that they are emergent behaviors. Through this exploration, we establish the model’s consistency with pathologies, providing valuable insights into the dynamic interplay of components influencing esophageal motility disorders. The results are obtained through parametric and robustness studies available in S4 Text. Robust in this context refers to the system’s ability to maintain its behavior despite perturbations.

Disordered non-occluding contractions.

The introduction of local variations to the parameters defining the neuronal circuitry results in sporadic or chaotic contraction patterns, as illustrated in Fig 6. Given the inherent variability of biological systems, small irregularities are expected. Introducing minor parameter perturbations—implemented as Gaussian variations with small standard deviations around the baseline values—does not substantially alter the contraction pattern, demonstrating the model’s robustness (see Fig F in S4 Text). However, when the magnitude of local parameter variations increases or when the mean value of a key parameter (e.g., e) deviates from its baseline (e.g., reduced from 15 to 12), the excitatory–inhibitory balance required for proper RACs is disrupted, and the coordinated antegrade contractions break down, giving rise to pronounced irregularities and spastic patterns (Fig 6).

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Fig 6. Spastic patterns obtained through a neuromechanical model mimicking esophageal FLIP test.

The figures display the color-coded spatio-temporal topographies of non-dimensional cross-sectional area with respect to A_o. is non-dimensional length with respect to L and is non-dimensional time with respect to . (A) Chaotic response created by irregularities in inhibitory neuronal pathways (e). (B) Chaotic response to irregularities in excitatory input from the distension-sensitive mechanoreceptors (w_E) with reduced inhibitory input to inhibitory pathways (see text). The parametric values used for these simulations are displayed in Tables H and I in S3 Text.

https://doi.org/10.1371/journal.pcbi.1013778.g006

It is crucial to emphasize the significance of the system’s robustness. If the system is not initially robust, it becomes more sensitive to small irregularities. Such cases have parametric values that allow the solution to exhibit regular RACs. Nevertheless, the excitatory-inhibitory balance is not as stable. Thus, these scenarios are more prone to sporadic or chaotic contraction patterns when introducing small irregularities, which would not otherwise trigger such responses (Fig 6B).

Sustained panesophageal contractions.

Sustained, non-propagating contractions leading to an increase in FLIP pressure are observed when the inhibitory signal to the excitatory population is weakened. Thus, inhibitory activity is insufficient to overcome excitatory activity, allowing excitatory cells to remain active. This scenario is reproduced by reducing the inhibitory signal to adjacent excitatory neurons (e) or by decreasing the excitation of inhibitory neurons (c or w_I). Note that since the FLIP’s fluid is incompressible, the color-coded topography of cross-sectional area may resemble absent contractile response (Fig 4A). However, the pressure profile differs, as the uniform muscle contraction increases the pressure in the bag, as illustrated in Fig 7.

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Fig 7. Internal bag pressure over time.

Non-dimensional internal bag pressure (with respect to K_e) over non-dimensional time (with respect to ) obtained in three distinct scenarios. Pressure is recorded at a single location along the bag length. In the case of normal RACs, the bag pressure fluctuates, reaching its peak during maximum contraction. Conversely, in absent contractile response, where no contraction occurs, the pressure remains consistently low throughout volumetric distension. In sustained panesophageal contraction, the entire esophagus contracts uniformly, increasing bag pressure that persists at a high level throughout the volumetric distension. The parametric values used to obtain these results are listed in Table 1 and Tables A and J in S3 Text.

https://doi.org/10.1371/journal.pcbi.1013778.g007

Repetitive retrograde contractions.

Repetitive retrograde contractions emerge with decreased excitation of inhibitory neurons via mechanoreceptors (S_I) or increased excitation of excitatory neurons via mechanoreceptors (S_E) (Fig 8). The retrograde pattern observed through our model is a result of a phase shift in the excitatory signal, opposing the antegrade pattern. Importantly, this phenomenon is not due to a retrograde traveling signal, as the excitatory and inhibitory pathways remain unidirectional. Further details on reversing propagation direction are discussed extensively in our recent work [53].

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Fig 8. A repetitive retrograde contractions pattern obtained through the neuromechanical model by disabling excitation of inhibitory neuronal population through mechanoreceptors (S_I).

The figure displays the spatio-temporal topography of the non-dimensional color-coded cross-sectional area with respect to A_o. is non-dimensional length with respect to L and is non-dimensional time with respect to . The parametric values used to obtain these results are listed in Table K in S3 Text.

https://doi.org/10.1371/journal.pcbi.1013778.g008

Overview of mathematical model

In this section, we provide a brief overview of esophageal neuromechanics and outline how its various components are represented in our empirically guided model. We refer to it as ‘empirically guided’ because the inclusion of specific neural and mechanical elements is mostly informed by clinical phenomenological experiments. The mathematical details of the model are discussed in the following sections.

The model focuses on the smooth muscle region of the esophagus, which underlies the response observed during FLIP testing. In this region, peristalsis can occur independently of central input and is primarily mediated by intrinsic enteric neural circuits that respond to local distension through mechanosensory feedback [10,33]. Thus, the model is constructed to represent the local, ENS-driven control of smooth muscle peristalsis without explicitly including extrinsic vagal or spinal pathways.

The esophagus undergoing FLIP testing is modeled as a closed-end, flexible tube filled with an incompressible, viscous fluid [81]. The circular muscle layer is modeled as the primary contractile element responsible for lumen closure and bolus propulsion. The longitudinal muscle layer is not explicitly included, as previous computational work demonstrated that circular muscle contraction alone is sufficient to generate effective bolus transport, whereas longitudinal muscle shortening alone cannot propel the bolus [82].

The neural circuitry is represented as a chain of coupled relaxation oscillators (Fig 2), a framework widely used to describe oscillatory or wave-like activity in biological systems, including central pattern generators [83], slow waves in the stomach [84], and intestines [24,85–88].

The coupling between the neuronal model and the body mechanics is modeled as follows. Since the esophageal response to FLIP is known to be involuntary and assumed to be triggered locally through distension-sensitive mechanoreceptors [39,42,89], the model is such that excitation of the circuit is led by local feedback. The neural circuitry receives excitatory signals from mechanoreceptors along the esophageal body (Fig 2A). In return, when the amount of local excitation rises above a given threshold, excitatory motor neurons cause a contraction of esophageal muscles [12,90]. The contraction wave is coordinated by a sequential excitation [32,58]. The innervation pathways for each neuronal population (excitatory and inhibitory) are discussed next.

Excitatory input from mechanoreceptors. When the FLIP bag inside the smooth muscle section of the esophagus is inflated, it causes the esophagus to distend, activating the proprioceptive channel that responds to changes in strain (deformation) [20,89]. These mechanoreceptors are part of the esophageal neural circuit and are distributed along the length of the esophagus. Thus, in our neuronal model, mechanoreceptors are activated when they sense that local strain increases above a threshold and send excitatory signals to both excitatory and inhibitory neuronal populations.

Since the FLIP records only from the smooth muscle portion of the esophagus, the present model focuses on ENS-driven mechanosensory feedback and does not explicitly include extrinsic vagal or spinal pathways. Thus, the model presumes all mechanoreceptors are entirely intrinsic to the ENS.

Since the esophagus contracts proximal to a bolus and relaxes distal to a bolus [31], activated mechanoreceptors are set to send excitatory signals to rostral excitatory populations and caudal inhibitory populations [10]. Additionally, it has been reported that during esophageal distension, the portion of the esophageal body that is located distal to the distended balloon typically remains inactive or quiescent [62]. The esophagus distal to the excited portion is inhibited to accommodate for the incoming bolus [10,31]. Therefore, the input from the distension-sensitive mechanoreceptors to the inhibitory neuronal population affects all inhibitory neurons caudal to the sensed distension. The input from mechanoreceptors to the excitatory neuronal population is more local, such that sensed distension only inputs an excitatory signal to rostral excitatory neurons that are within 10% of esophageal length.

Note that while our model includes only intrinsic mechanoreceptors responsive to distension, these can be conceptualized as components of a broader mechanosensory network that also includes extrinsic vagal and spinal afferents.

Input from adjacent neuronal populations.

Compared with the small intestine and colon, the organization of intrinsic interneuron-like elements in the esophageal myenteric plexus is less well defined in the literature [9,22,66,91,92]. Nevertheless, descending neural pathways have been described in which neurons project orally to synapse onto cholinergic excitatory motor neurons, thereby initiating contraction in response to esophageal distention [93]. In addition, depolarization of one muscle cell in the esophagus can induce electrotonic depolarization of neighboring cells in an aboral direction, contributing to the coordinated propagation of motor activity [32,94].

To capture these coordinated effects with a compact segmental model, nearest-neighbor couplings are introduced as phenomenological reductions of multiple biological processes [95–97]. A unidirectional aboral excitatory coupling between excitatory populations of adjacent segments (parameter b in Fig 2A) reproduces the dominant directionality of esophageal propagation and abstracts the combined effects of descending excitatory projections. A directed inhibitory coupling from the inhibitory population in one segment to the caudal excitatory population (parameter d in Fig 2A) captures the functional consequence of inhibitory reflex signaling that shapes wavefront timing and spatial patterning.

Intrasegmental synaptic inputs. Within each segment, local interactions between excitatory and inhibitory neuronal populations determine the overall excitatory drive to motor output. In the model, these interactions are represented by a set of coupling terms (parameters a, c, e, and f in Fig 2A). These are trivial elements that capture how activity within a segment regulates its own excitability and responsiveness, and are often used when modeling the electrical activity using a phenomenological approach [87,95,98]. They are not intended to reproduce detailed synaptic anatomy.

Note that there is no direct evidence for direct inhibitory synapses within the esophageal myenteric plexus. However, functional inhibitory signaling in the esophagus is well documented and is important for coordinating contraction-relaxation sequences [14,93]. In the model, this inhibitory influence is represented as an inhibitory input to the excitatory motor population, which effectively produces muscle relaxation by suppressing excitatory drive rather than by introducing an explicit inhibitory motor-to-muscle projection.

Table A in S2 Text provides additional information on the neural circuit.

Body mechanics and fluid equations

The flow inside the FLIP device placed in the esophageal lumen is modeled as a one-dimensional, fluid-filled, flexible tube that is closed on both ends [81,99,100]. The conservation of mass and momentum equations are

(1)

and

(2)

respectively. A parabolic flow is assumed everywhere. In the equations above, A(x,t), u(x,t), P(x,t), and are the tube cross-sectional area, fluid velocity (averaged at each cross-sectional area), pressure inside the tube, fluid density, and fluid viscosity, respectively. We introduce a linear constitutive relation

(3)

to complete the system [101,102]. Here, P_o, K_e, and A_o are external pressure, tube stiffness, and undeformed reference area (cross-sectional area of the tube when P = P_o), respectively. Lastly, the neurally controlled contraction of the esophageal lumen is set through dynamically varying the rest cross-sectional area of the tube by a factor [103,104]. captures the muscular dynamics, induced by the neural activity discussed next.

Neural circuitry: excitation, inhibition, and muscle contraction pattern

To represent the electrical activity of the entire organ, we introduce a system of locally coupled Wilson–Cowan oscillators, distributed uniformly along the length of the esophagus. Each unit consists of two neuronal populations, excitatory (E) and inhibitory (I) (Fig 2A). The values of E_i(t) and I_i(t) at each oscillator depict the activity levels of the excitatory and inhibitory neuron populations. The differential equations for the time-dependent variation of averaged excitatory and inhibitory neuronal activities at node i introduced by [51] with the unidirectional coupling introduced by [97] are

(4)

and

(5)

The intrasegmental connectivity parameters a, e, c, and f represent the average synaptic weight of excitatory and inhibitory synapses per cell in the excitatory or inhibitory population. The connectivity coefficient b denotes the unidirectional excitatory connections between the excitatory populations of neighboring segments (excitatory signal from nearest posterior excitatory population). The connectivity coefficient d signifies the unidirectional inhibitory connections from a segment’s inhibitory population to the nearest anterior segment’s excitatory population. For the relaxation oscillators located at the rostral end, b = d = 0. The time constants and dictate the decay of the excitatory and inhibitory activities, and determine the timescales of the activities. The sigmoid function characterizes the switching threshold defined as

(6)

where is activation speed (slope of sigmoid), and is activation threshold (location of sigmoid’s maximum slope) [96,97]. Lastly, S_E,i and S_I,i symbolize the local excitatory inputs to each population at oscillator i, capturing the mechano-sensory feedback coming from the distension-sensitive mechanoreceptors [14,15], discussed next.

The distension-sensitive mechanoreceptors are activated when they sense that local deformations increase above a threshold (). In our model, it is equivalent to pressure or hoop stress dependent trigger defined by . By including , we ensure that the receptors sense strain relative to an updated reference area rather than the rest reference area (A_o). As discussed in a previous section, when activated, a mechanosensory receptor sends excitatory signals to rostral excitatory populations and excitatory signals to caudal inhibitory populations. The input to excitatory neurons from distension-sensitive mechanoreceptors is more local, meaning that distension excites excitatory neurons just proximal of the activated receptor. On the other hand, the input from distension-sensitive mechanoreceptors to inhibitory neurons is applied to all inhibitory neurons caudal to the activated receptor. Thus, S_E and S_I input for a segment located in x_i are defined as

(7)

where , w_E and w_I are the strength of the net sensory feedback coming from the distension-sensitive mechanoreceptors receptors, , , , L is esophageal length, g_S and g_E are gains, and x_s is horizontal shift. The schematic of the distension-induced excitation is presented in Fig 2B.

When the excitatory neural activity surpasses a certain threshold, it excites the smooth muscles, causing them to contract. The coupling between the neural circuit and the mechanical model is done through solving for the activation function , which receives input from excitatory neuronal population E, such that

(8)

In this equation, is a time constant, the excitatory threshold for muscle activation, and , where is the gain, and is the maximum contraction strength.

In the muscle pattern equation above, only excitatory motor pathways are represented, providing a minimal framework to examine how mechanosensory feedback within the ENS can generate and sustain secondary peristaltic activity. Incorporating inhibitory motor pathways, as previously done in [24] and [85], would allow for more complete reflex dynamics, but is beyond the present scope.

Non-dimensional form

The dynamic system of equations described above is non-dimensionalized using

(9)

where , , U, p, and are the non-dimensional variables for cross-sectional area, time, velocity, pressure, and position, respectively [81,99]. Thus, dimensional Eqs ((1), (2), (3), (4), (5), and (8)) become

(10)

(11)

(12)

(13)

(14)

(15)

respectively, where , , , and .

Initial and boundary conditions

The fluid in the tube is initially at rest and the cross-sectional area is uniform, hence

(16)

where [100,105]. Additionally,

(17)

since the neuronal system is at rest. Lastly, recall that the tube is closed on both ends, thus,

(18)

To obtain a boundary condition for , we plug Eq (12) and the velocity boundary condition in Eq (18) into Eq (11), which yields

(19)

Numerical implementation

The system of equations is solved using MATLAB ode45 function for the time derivatives and central difference discretization for the spatial derivatives. The discrete expressions for the external source terms in Eq (7) are

(20)

where N denotes the number of nodes.

Table 1 lists the model’s parameters and their values. Values for the parameters defining esophageal body mechanics and fluid properties are approximated based on clinical data and previous computational studies [81,82,99,102]. The parameters involved in the neural equations are chosen such that the system has a single unstable fixed point and a stable limit cycle in response to the constant external stimulus S_E and S_I. Depending on the values of the parameters in the system, one can obtain multiple equilibria with different stability properties. For simplicity, we choose the values introduced by [51] with small variations. For example, theoretical work concerning Wilson-Cowan oscillators often sets S_I = 0 for simplicity [51]. By setting , we ensure that both excitatory and inhibitory populations “turn-on” when distension is applied, independent of one another. A theoretical overview of relaxation oscillators, limit cycle solution, and the constraints introduced by modeling esophageal response is provided in the S1 Text.

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Table 1. List of parameters and their values.

https://doi.org/10.1371/journal.pcbi.1013778.t001