Reduced computational modelling of Kölliker‐Fuse contributions to breathing patterns in Rett syndrome

Abstract

Key points

• Reduced computational models are used to test effects of loss of inhibition to the Kölliker‐Fuse nucleus (KFn).

• Three reduced computational models that simulate eupnoeic and vagotomized respiratory rhythms are considered.

• All models exhibit the emergence of respiratory perturbations associated with Rett syndrome as inhibition to the KFn is diminished.

• Simulations suggest that application of 5‐HT_1A agonists can mitigate the respiratory pathology.

• The three models can be distinguished and tested based on their predictions about connections and dynamics within the respiratory circuit and about effects of perturbations on certain respiratory neuron populations.

Abstract

Rett syndrome (RTT) is a developmental disorder that can lead to respiratory disturbances featuring prolonged apnoeas of variable durations. Determining the mechanisms underlying these effects at the level of respiratory neural circuits would have significant implications for treatment efforts and would also enhance our understanding of respiratory rhythm generation and control. While experimental studies have suggested possible factors contributing to the respiratory patterns of RTT, we take a novel computational approach to the investigation of RTT, which allows for direct manipulation of selected system parameters and testing of specific hypotheses. Specifically, we present three reduced computational models, developed using an established framework, all of which successfully simulate respiratory outputs across eupnoeic and vagotomized conditions. All three models show that loss of inhibition to the Kölliker‐Fuse nucleus reproduces the key respiratory alterations associated with RTT and, as suggested experimentally, that effects of 5‐HT_1A agonists on the respiratory neural circuit suffice to alleviate this respiratory pathology. Each of the models makes distinct predictions regarding the neuronal populations and interactions underlying these effects, suggesting natural directions for future experimental testing.

Key points

• Reduced computational models are used to test effects of loss of inhibition to the Kölliker‐Fuse nucleus (KFn).

• Three reduced computational models that simulate eupnoeic and vagotomized respiratory rhythms are considered.

• All models exhibit the emergence of respiratory perturbations associated with Rett syndrome as inhibition to the KFn is diminished.

• Simulations suggest that application of 5‐HT_1A agonists can mitigate the respiratory pathology.

• The three models can be distinguished and tested based on their predictions about connections and dynamics within the respiratory circuit and about effects of perturbations on certain respiratory neuron populations.

Introduction

Rett syndrome (RTT) is a developmental disorder that is caused by mutations in the X‐linked gene MECP2. Most RTT patients are young females; males with RTT typically do not survive the first years of life, although there are rare cases where affected males survive infancy. The disorder is responsible for a variety of symptoms, including neurocognitive impairment and pyramidal and extrapyramidal dysfunction. One of the most dangerous and disruptive symptoms of RTT is dysfunctional autonomic regulation of breathing, which is characterized by frequent and random apnoeas, periodic breathing (alternating bouts of rapid and low frequency breathing), and breath holds. Many instances of sudden death of children with RTT are a result of respiratory distress (Weese‐Mayer et al. 2006).

The MECP2 gene encodes the methyl‐CpG binding protein 2 (MECP2), a transcriptional regulator that plays a role in modulating the expression of a variety of neurotransmitters, neuromodulators, receptors and neurotrophic factors. There has been a wealth of research using Mecp2 knockout mice (KO mice) to characterize RTT respiratory dysfunction (Stettner et al. 2007), to examine physiological consequences of MECP2 deficiency, and to investigate possible clinical approaches to address associated concerns. Among the many neurotransmitters, neuromodulators and neurotrophins found to be deficient in KO mice are GABA (Chao et al. 2010), noradrenaline (norepinephrine; Viemari et al. 2005), serotonin (Viemari et al. 2005), and BDNF (Kline et al. 2010). Furthermore, a variety of studies have found that acute and/or systematic application of various deficient neurotransmitters or selective receptor agonists can successfully rescue normal respiratory function to various degrees (Viemari et al. 2005; Abdala et al. 2010, 2014 a, b ; Kline et al. 2010; Levitt et al. 2013; Kron et al. 2014). Of particular interest are 5‐HT_1A agonists, which are currently being clinically tested for the specific indication of correcting breathing abnormalities in RTT (ClinicalTrials.gov).

Decades of research has established that the respiratory central pattern generator (rCPG) includes rhythmically interacting populations of inhibitory and excitatory neurons in the ventral respiratory column (VRC) in the medulla. Neurons within the VRC that activate at similar phases within the respiratory cycle tend to be spatially co‐localized. Inspiratory and expiratory neurons are predominantly concentrated in the pre‐Bötzinger (pre‐BötC) and Bötzinger (BötC) complexes, respectively, although this compartmentalization is not precise (Lindsey et al. 2012; Ramirez & Baertsch, 2018). Descending projections from the VRC target premotor neurons in the rostral and caudal ventral respiratory groups (rVRG and cVRG) (Smith et al. 2013). The rCPG circuits are subject to tonic and phasic drives from a variety of sources, including: (i) chemoreception mediated through the retrotrapezoid nucleus/parafacial respiratory group (RTN/pFRG) and caudal raphe; (ii) mechanoreceptive feedback from pulmonary stretch receptors and chemoreceptive input from carotid bodies mediated through the nucleus solitarus (NTS); and (iii) input from various regions of the pons, perhaps most significantly the lateral parabrachial complex, including the Kölliker‐Fuse nucleus (KFn) (Kubin et al. 2006; Dutschmann & Dick, 2012; Smith et al. 2013). In normal breathing (or eupnoea), afferent feedback contributes to regulating the transition between inspiration and expiration (Kubin et al. 2006). However, it has become clear that the pons (particularly the KFn) also plays a major role in gating this transition, capable of acting in the absence of afferent feedback to help maintain a normal rhythm (Dick et al. 2008; Dutschmann & Dick, 2012). It is clear that RTT involves imbalances in this complex network of interacting populations. Recent research suggests that a major cause of RTT respiratory arrhythmia is dysfunction in the interaction between NTS‐mediated afferent feedback and pontine activity, causing overexcitability of KF neurons and dysregulation of the inspiratory off‐switch (IOS). This dysfunctional interaction is hypothesized to result from deficiency in GABAergic and/or serotonin‐mediated inhibition (Stettner et al. 2007; Dutschmann & Dick, 2012; Abdala et al. 2014a, 2014b, 2016).

There is a long tradition of using computational techniques to build models of this complex system, develop intuition about how it functions, and test hypotheses about the generation of respiratory rhythms (Lindsey et al. 2012; Molkov et al. 2017). These models exist in many forms, from large scale models (Rybak et al. 2008) to reduced population activity‐based models (Rubin et al. 2009), and have been designed to focus on a variety of specific aspects of respiratory function, including breathing following brainstem transections (Smith et al. 2007), active expiration (Rubin et al. 2011; Molkov et al. 2014), and interacting pontine and afferent regulation of the breathing cycle (Molkov et al. 2013). Until now, however, there has not been a model designed to investigate RTT respiratory arrhythmia. In this work, we present three such models, using a reduced, population activity‐based framework that allows us to identify the mechanisms underlying altered dynamics resulting from changes based on experimental results from KO mice. The models presented here were designed to examine the hypothesis that reduction or elimination of inhibition from medullary and pontine respiratory populations to the KF nucleus can result in the onset of RTT‐like respiratory arrhythmia (Abdala et al. 2016). A variety of other well‐characterized respiratory patterns are explored, in order to validate these models and thus establish their suitability for examination of simulated RTT conditions. Finally, the models are used to explore a mechanistic rationale for experimental findings using 5‐HT_1A agonists to treat RTT respiratory dysfunction (Abdala et al. 2010, 2014 a, b ). Based on these results, the models considered yield predictions related to the network dynamics underlying the emergence and potential suppression of RTT respiratory dysfunction, particularly spontaneous apnoeas. These predictions include some non‐intuitive mechanistic insights and suggest some new directions for experimental investigation.

Methods

We developed, simulated, and analysed three reduced respiratory models. Each model was considered in the following regimes: eupnoea, Rett syndrome (RTT), and RTT with 5‐HT_1A agonist application, all in intact and in vagotomized cases.

Model structure

One of the reduced models contains three distinct respiratory neuronal populations, and we refer to it as the three population (3p) model. The other two models contain four distinct respiratory neuronal populations each and have identical structure but different parameter tunings; we refer to these as the four population escape (4p‐e) model and the four population release (4p‐r) models. These names derive from the mechanism by which apnoea is terminated in RTT simulations; details are discussed below. The general structure of the models is inspired by the models used in several earlier respiratory modelling studies (Rubin et al. 2009, 2011; Molkov et al. 2014). The equations in this paper are based on one of these (Rubin et al. 2011), except that rather than incorporating instantaneous synapses, we use the time‐dependent synaptic variable dynamics from an earlier work (Daun et al. 2009). Schematic diagrams illustrating the components of these models appear in Fig. 1. Neuronal populations are considered to be synchronized in terms of transitions between active spiking and silent phases, although not in terms of precise spike times, and are thus effectively represented by single non‐spiking neuron equations. The potential of each single neuron represents the average membrane potential of the corresponding population. The models that we consider reduce the four‐neuron medullary respiratory kernel utilized in the previous studies into two neurons: a pre‐Bötzinger complex (pre‐BötC) neuron and a Bötzinger complex (BötC) neuron. The pre‐BötC component is used to represent inspiratory pre‐motor output, while the BötC population is used to represent the cumulative expiratory pre‐motor output. Thus, for the sake of simplicity, ‘I population’, ‘inspiratory population’, and ‘pre‐BötC’ are used interchangeably, as are ‘E population’, ‘expiratory population’, and ‘BötC’.

Figure 1. Schematic illustrations of model components.

A, the 3p model includes three populations of potentially active respiratory neurons, representing neurons from the pre‐BötC, BötC and Kölliker‐Fuse (KF‐E). It also features medullary and pontine sources of tonic excitatory synaptic drive (green triangles). The model includes a simplified representation of pulmonary stretch receptor (PSR) feedback related to pre‐BötC output and mediated through the nucleus solitarius (NTS). B, the 4p‐r and 4p‐e models share the same structure, which includes the same populations as the 3p model along with an additional parabrachial (PB‐I) population. Red circles: active (or more active) during inspiration; blue circles: active (or suppressed) during expiration; connections capped with arrows: excitatory synaptic pathways; connections capped with circles: inhibitory synaptic pathways; V: cut to represent vagotomy; R: altered to represent RTT; 5‐HT: altered as part of the representations of 5‐HT_1A agonist application; yellow boxes and dashed lines: components and pathways that are collapsed into a simplified drive rather than modelled separately. [Color figure can be viewed at wileyonlinelibrary.com]

The medullary kernel is modulated by a variety of inputs. As in previous studies (Rubin et al. 2009, 2011; Molkov et al. 2014), some of these inputs are represented as simple tonic drives stemming from tonic populations in the medulla and the pons. The models contain a simplified representation of pulmonary stretch receptor (PSR) feedback (as mediated through the NTS) derived from the output of the I population, in contrast to explicit representations of the lungs and NTS pump cells (Molkov et al. 2014). Additionally, a novel feature in the two 4p models is the inclusion of two phasic pontine populations: an inspiratory‐modulating parabrachial population (PB‐I) and an expiratory‐modulating Kölliker‐Fuse (KF) population (KF‐E). Alternatively, in the 3p model, we include only one phasic pontine population, representing the KF‐E. The connections involving the pontine populations are based on a previous large scale model (Rybak et al. 2008) and experimental evidence of intrapontine inhibition (Cohen, 1971; Okazaki et al. 2002; Morschel & Dutschmann, 2009; Dutschmann & Dick, 2012), including studies supporting the idea that KF activity induces inhibitory effects (see Dutschmann & Dick, 2012 and the references therein).

In all models, vagotomy is simulated by deactivating all pathways mediated by the NTS. The models do not mathematically represent peripheral chemoreceptors. RTT is simulated by weakening GABAergic inhibition of the KF‐E population. Recent findings have demonstrated that blocking GABAergic inhibition in the KF of wild‐type rats is sufficient to create an RTT‐like respiratory pattern (Abdala et al. 2016). Thus, in RTT simulations, we only weaken inhibitory projections to the KF from populations proven to release GABA, namely the NTS, BötC and PB‐I (Ezure et al. 2003, 2003; Ezure & Tanaka, 2004). We also explore the use of a 5‐HT_1A agonist as a treatment for RTT respiratory dysfunction (Levitt et al. 2013; Abdala et al. 2014 a, b ). To simulate the application of a 5‐HT_1A agonist, we strengthened inhibitory connections within the medullary kernel and we activated 5‐HT_1A‐sensitive potassium channels in all populations (detailed in the equations below), based on previous modelling work (Shevtsova et al. 2011).

All model neurons include an intrinsic burst generation capability within some parameter regime based on the activity of the persistent sodium currrent (INaP ) (Butera et al. 1999; Del Negro et al. 2002; Koizumi et al. 2008; Daun et al. 2009) (see Table 1 below), but in practice most are not tuned to the intrinsic bursting regime and their outputs are largely influenced by connections within the networks shown in Fig. 1 (Smith et al. 2007).

Table 1.

Model neuron intrinsic behaviours

		3p	4p‐r	4p‐e
Tonic drive intact	Pre‐BötC	Tonic at V = −37 mV	Tonic at V = −36 mV	Tonic at V = −35 mV
	BötC	Oscillatory	Tonic at V = −39 mV	Tonic at V = −37 mV
	KF‐E	Tonic at V = −30 mV	Tonic at V = −35 mV	Tonic at V = −33 mV
	KF‐I	—	Tonic at V = −38 mV	Tonic at V = −38 mV
Tonic drive removed	Pre‐BötC	Oscillatory	Oscillatory	Oscillatory
	BötC	Quiescent at V = −60 mV	Oscillatory	Oscillatory
	KF‐E	Tonic at V = −30 mV	Tonic at V = −35 mV	Oscillatory
	KF‐I	—	Oscillatory	Oscillatory

Model specification

In describing the 4p models, we use subscripts i ∈ {pbc,bc,kf‐e,pb‐i} to represent the I (pre‐BötC), E (BötC), KF‐E and PB‐I populations, respectively. In the equations for the 3p model, we use i ∈ {pbc,bc,kf‐e} to represent the I (pre‐BötC), E (BötC), and KF‐E populations, respectively. The average membrane potential of each population in all models evolves according to the voltage ordinary differential equation (ODE):

$$\begin{array}{ccc}& & \frac{\mathrm{d}{V}_i}{\mathrm{d}t}=(-I_{\mathrm{Na}{\mathrm{P}}_i}-I_{{\mathrm{K}}_i}-I_{{\mathrm{L}}_i}-I_{\mathrm{to}{\mathrm{n}}_i}\hfill \\ & & -I_{\mathrm{syn}-{\mathrm{I}}_i}-I_{\mathrm{syn}-{\mathrm{E}}_{\mathrm{i}}}-I_{\mathrm{K}{\mathrm{S}}_i}-{\mathrm{\Gamma }}_i)/C.\hfill \end{array}$$ (1)

In eqn (1), $I_{{\mathrm{NaP}}_i}$ represents the current through persistent sodium channels, $I_{{\mathrm{K}}_i}$ is the potassium delayed rectifier current, $I_{{\mathrm{L}}_i}$ is the leakage current, $I_{{\mathrm{ton}}_i}$ is cumulative current from all sources of tonic drive, and are currents from inhibitory and excitatory synaptic channels, respectively, $I_{{\mathrm{KS}}_i}$ is current through 5‐HT_1A‐sensitive potassium channels, Γ_i is a noise term and C is the membrane capacitance. The transmembrane currents are given by the following equations:

$$\begin{array}{ccc}& & I_{\mathrm{Na}{\mathrm{P}}_i}=g_{\mathrm{Na}{\mathrm{P}}_i}{m}_{\mathrm{NaP},\infty }\left(V_i\right)h_i\left(V_i-E_{\mathrm{Na}}\right),\hfill \\ & & I_{{\mathrm{K}}_i}=g_{\mathrm{K}}{m}_{\mathrm{K},\infty }{(V_i)}^4\left(V_i-E_{\mathrm{K}}\right),\hfill \\ & & I_{{\mathrm{L}}_i}=g_{\mathrm{L}}\left(V_i-E_{\mathrm{L}}\right),\hfill \\ & & I_{\mathrm{to}{\mathrm{n}}_i}=c_i\left(V_i-E_{\mathrm{syn}-\mathrm{E}}\right),\hfill \\ & & {\displaystyle I_{\mathrm{syn}-{\mathrm{I}}_i}=g_{\mathrm{syn}-\mathrm{I}}\left(V_i-E_{\mathrm{syn}-\mathrm{I}}\right)\left(1+k_s{k}_{\alpha 3_i}\right)\sum _{j=1;j\ne i}^4{b}_{j,i}{s}_{j,i},}\hfill \\ & & {\displaystyle I_{\mathrm{syn}-{\mathrm{E}}_i}=g_{\mathrm{syn}-\mathrm{E}}\left(V_i-E_{\mathrm{syn}-\mathrm{E}}\right)\sum _{j=1;j\ne i}^4{a}_{j,i}{s}_{j,i},}\hfill \\ & & I_{\mathrm{K}{\mathrm{S}}_i}=g_{\mathrm{K}{\mathrm{S}}_i}{m}_{\mathrm{KS}}\left(\left[S\right]\right)\left(V_i-E_{\mathrm{K}}\right),\hfill \end{array}$$ (2)

while Γ_i is a normal random variable with zero mean and standard deviation γ_i.

Inactivation of persistent sodium channels (h_i) and the time course of the synaptic conductance from cell j to cell i (s_j,i) are modelled using the following ODEs:

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}{h}_i}{\mathrm{d}t}& =& (h_{{\infty }_i}(V_i)-h_i)/{\tau }_{h_i}(V_i),\hfill \\ \hfill \frac{\mathrm{d}{s}_{j,i}}{\mathrm{d}t}& =& \alpha (1-s_{j,i})s_{{\infty }_i}(V_j)-{\beta }_{j,i}{s}_{j,i},\hfill \end{array}$$ (3)

which are standard in the Hodgkin‐Huxley framework.

Voltage‐dependent activation functions and time constants in eqn set (3) are described by the following functions:

$$\begin{array}{c}{h}_{{\infty }_i}(V_i)=1/(1+\exp [(V_i-{\theta }_{h_i})/{\sigma }_h]),\hfill \\ s_{{\infty }_i}(V_i)=1/(1+\exp [(V_i-{\theta }_{\mathrm{sy}{\mathrm{n}}_{j,i}})/{\sigma }_{\mathrm{sy}{\mathrm{n}}_{j,i}}]),\hfill \\ {\tau }_{h_i}(V_i)={\epsilon }_i/(\cosh [(V_i-{\theta }_{{\tau }_i})/{\sigma }_{h_i}/2]),\hfill \\ m_{\mathrm{NaP},\infty }(V_i)=1/(1+\exp [(V_i-{\theta }_{m_{\mathrm{NaP}}})/{\sigma }_{m_{\mathrm{NaP}}}]),\hfill \\ m_{\mathrm{K},\infty }(V_i)=1/(1+\exp [(V_i-{\theta }_{m_{\mathrm{K}}})/{\sigma }_{m_{\mathrm{K}}}]).\hfill \end{array}$$ (4)

Note that in the s _∞(V) equation in eqn set (4), the subscript j denotes the source or presynaptic population, the subscript i the post‐synaptic target.

Application of a 5‐HT_1A agonist is simulated with two changes in the model: increased inhibitory strength and activation of I _KS channels (Levitt et al. 2013). The first perturbation is implemented by making the factor k_s in the equation in eqn set (2) non‐zero, and the second depends on the following function that appears in eqn set (2), where S is the agonist concentration and S _t is a saturation threshold (Shevtsova et al. 2011):

$$m_{\mathrm{KS}}\left(\left[S\right]\right)=1/(1+exp[({[S]}_{\mathrm{t}}-[S])/2]).$$ (5)

Note that I _KS has a straightforward hyperpolarizing effect and hence could also represent contributions from other sources of hyperpolarization with similar reversal potentials (cf. Manzke et al. 2010; Rojas & Fiedler, 2016).

The tonic drive term $I_{{\mathrm{ton}}_i}$ includes a coefficient c_i reflecting the inclusion of two sources, the pons and the medulla, the contributions of which are represented by ρ_i and μ_i, respectively:

$$c_i={\rho }_i+{\mu }_i.$$ (6)

Table 1 describes the intrinsic behaviour of the neurons in all three models, both with and without the tonic excitatory connections (green triangles in Fig. 1). Knowing how each model neuron behaves in the absence of phasic inputs from other neurons can provide important context for the model dynamics and predictions. Table 2 lists the values of parameters that are not population specific and are shared by all models. Tables 3 and 4 present the values of additional non‐synaptic parameters and synaptic parameters, respectively, in the 3p model; Tables 5 and 6 display the values of these non‐synaptic and synaptic parameters, respectively, in the 4p‐r model; and Tables 7 and 8 include the values of these non‐synaptic and synaptic parameters in the 4p‐e model. Table 9 indicates parameter changes implemented to simulate RTT. Parameter values for all models were derived by starting from previous models (Rubin et al. 2011; Shevtsova et al. 2011; Molkov et al. 2014) and introducing perturbations to achieve appropriate dynamics. In the 3p model, the KF‐E unit could initiate the inspiratory off‐switch in the vagotomized state (Poon & Song, 2014; Song et al. 2015). In the 4p models, the introduction of the PB‐I population resulted in an intact source of inhibition to the KF‐E population during the inspiratory phase of the vagotomized regime. Therefore, in the 4p models, KF‐E could not directly initiate the I‐E phase transition during vagotomy, as occurs in the 3p model, but nonetheless contributed to expiration in the vagotomized state (Morschel & Dutschmann, 2009).

Table 2.

Shared parameters

Conductances (nS)	Reversal potentials (mV)	Slope factors (mV)	Other
g L = 2.8	E L = −65.0	σh = 6.0	C = 21.0 pF
g syn‐E = 10.0	E Na = 50.0	σm = −6.0	[S]t = 0 or 10 μM
g syn‐I = 60.0	E K = −85.0		α = 1
	E syn‐E = 0.0
	E syn‐I = −80.0

Table 3.

Non‐synaptic 3p parameters

Conductances (nS)		Half‐activations (mV)		Time constants (ms)	Noise deviations (nA)	5‐HT1A scaling factors
$g_{\mathrm{Na}{\mathrm{P}}_{\mathrm{pbc}}}=10.0$	$g_{\mathrm{K}{\mathrm{S}}_{\mathrm{pbc}}}=1.0$	${\theta }_{h_{\mathrm{pbc}}}=-50$	${\theta }_{{\tau }_{\mathrm{pbc}}}=-35$	ϵpbc = 400	γpbc = 0.5	$k_{\alpha 3_{\mathrm{pbc}}}=0.35$
$g_{\mathrm{Na}{\mathrm{P}}_{\mathrm{pbc}}}=5.0$	$g_{\mathrm{K}{\mathrm{S}}_{\mathrm{bc}}}=0.3$	${\theta }_{h_{\mathrm{bc}}}=-50$	${\theta }_{{\tau }_{\mathrm{bc}}}=-25$	ϵbc = 1000	γbc = 1	$k_{\alpha 3_{\mathrm{bc}}}=1$
				ϵkf‐e = 1750	γkf‐e = 1
g K = 0		θm = −37

Table 4.

Synaptic 3p parameters

		Pre‐BötC	BötC	KF‐E
Inhibitory synaptic scaling factors	Pre‐BötC	—	b pbc , bc = 0.0417	b pbc , kf‐e = 0.0333
	BötC	b bc , pbc = 0.0417	—	b bc , kf‐e = 0.0083
Excitatory synaptic scaling factors	Pre‐BötC	—	a pbc , bc = 0.03	—
	BötC	—	a kf‐e , bc = 0.03	—
Excitatory tonic scaling factors	Pons	ρpbc = 0.55	ρbc = 0.11	ρkf‐e = 0
	Medulla	μpbc = 0.45	μbc = 0.09	μkf‐e = 0
Synaptic half‐activations (mV)		${\theta }_{{\mathrm{syn}}_{j,i}}=-35$ for all other j,i
Synaptic slopes		${\theta }_{\mathrm{sy}{\mathrm{n}}_{j,i}}=-3$ for all other j,i
Synaptic decay rates	βpbc,bc = 0.005	βpbc,kf‐e = 0.005	βj,i = 0.08 for all other j,i

Table 5.

Non‐synaptic 4p‐r parameters

Conductances (nS)		Half‐activations slope factors (mV)		Time constants (ms)	Noise deviations (nA)	5‐HT1A scaling factors
$g_{\mathrm{Na}{\mathrm{P}}_{\mathrm{pbc}}}=10.0$	$g_{\mathrm{K}{\mathrm{S}}_{\mathrm{pbc}}}=1.0$	${\theta }_{h_{\mathrm{pbc}}}=-47.0$	${\theta }_{{\tau }_{\mathrm{pbc}}}=-35.0$	ϵpbc = 600	γpbc = 1	$k_{\alpha 3_{\mathrm{pbc}}}=1$
$g_{\mathrm{Na}{\mathrm{P}}_{\mathrm{bc}}}=5.0$	$g_{\mathrm{K}{\mathrm{S}}_{\mathrm{bc}}}=1.0$	${\theta }_{h_{\mathrm{bc}}}=-47.0$	${\theta }_{{\tau }_{\mathrm{bc}}}=-35.0$	ϵbc = 700	γbc = 1	$k_{\alpha 3_{\mathrm{bc}}}=1$
				ϵkf‐e = 1500	γkf‐e = 1
				ϵpb‐i = 300	γpb‐i = 1
g K = 3.0		θm = −40.0	${\theta }_{m_{\mathrm{K}}}=-30.0$
			${\sigma }_{m_{\mathrm{K}}}=-4.0$

Table 6.

Synaptic 4p‐r parameters

		Pre‐BötC	BötC	KF‐E	PB‐I
Inhibitory synaptic scaling factors	Pre‐BötC	—	b pbc , bc = 0.03	b pbc , kf‐ e = 0.1	—
	BötC	b bc , pbc = 0.035	—	b bc , kf‐e = 0.001	—
	KF‐E	—	—	—	b kf‐e,pb‐i = 0.05
	PB‐I	—	—	b pb‐i,kf‐e = 0.05	—
Excitatory synaptic scaling factors	Pre‐BötC	—	a pbc , bc = 0.03	—	a pbc , pb‐i = 0.15
Excitatory tonic scaling factors	Pons	ρpbc = 0.15	ρbc = 0.5	ρkf‐e = 0	ρpb‐i = 0.8
	Medulla	μpbc = 0.15	μbc = 0.15	μkf‐e = 0	μpb‐i = 0
Synaptic half‐activations (mV)				${\theta }_{\mathrm{sy}{\mathrm{n}}_{j,i}}=-35$ for all other j,i
Synaptic slopes				${\sigma }_{\mathrm{sy}{\mathrm{n}}_{j,i}}=-3$ for all other j,i
Synaptic decay rates		βpbc , bc = 0.005	βpbc , kf‐e = 0.005	βj,i = 0.08 for all other j,i

Table 7.

Non‐synaptic 4p‐e parameters

Conductances (nS)		Half‐activations/slope factors (mV)		Time constants (ms)	Noise deviations (nA)	5‐HT1A scaling factors
$g_{\mathrm{Na}{\mathrm{P}}_{\mathrm{pbc}}}=10.0$	$g_{\mathrm{K}{\mathrm{S}}_{\mathrm{pbc}}}=0.6$	${\theta }_{h_{\mathrm{pbc}}}=-47.0$	${\theta }_{{\tau }_{\mathrm{pbc}}}=-35.0$	ϵpbc = 600	γpbc = 1	$k_{\alpha 3_{\mathrm{pbc}}}=1$
$g_{\mathrm{Na}{\mathrm{P}}_{\mathrm{bc}}}=5.0$	$g_{\mathrm{K}{\mathrm{S}}_{\mathrm{bc}}}=1.2$	${\theta }_{h_{\mathrm{bc}}}=-47.0$	${\theta }_{{\tau }_{\mathrm{bc}}}=-35.0$	ϵbc = 800	γbc = 1	$k_{\alpha 3_{\mathrm{bc}}}=1$
				ϵkf‐e = 1500	γkf‐e = 1
				ϵpb‐i = 300	γpb‐i = 1
g K = 3.0		θm = −40.0	${\theta }_{m_{\mathrm{K}}}=-30.0$
			${\sigma }_{m_{\mathrm{K}}}=-4.0$

Table 8.

Synaptic 4p‐e parameters

		Pre‐BötC	BötC	KF‐E	PB‐I
Inhibitory synaptic scaling factors	Pre‐BötC	—	b pbc , bc = 0.03		—
	BötC	b bc , pbc = 0.0345	—		—
	PB‐I	—	—		—
Excitatory synaptic scaling factors	Pre‐BötC	—	a pbc , bc = 0.03	—
	KF‐E	—		—	—
Excitatory tonic scaling factors	Pons	ρpbc = 0	ρbc = 0.35
	Medulla	μpbc = 0.3	μbc = 0.35
Synaptic half‐activations (mV)				${\theta }_{\mathrm{sy}{\mathrm{n}}_{j,i}}=-35$ for all other j,i
Synaptic slope factors (mV)				${\sigma }_{\mathrm{sy}{\mathrm{n}}_{j,i}}=-3$ for all other j,i
Synaptic decay rates		βpbc , bc = 0.005		βj,i = 0.08 for all other j,i

Table 9.

RTT parameters

		b pbc,kf‐e	b bc,kf‐e	b pb‐ i ,kf‐ e
3p	Intact	0.0333	0.0083	—
	Mod RTT	0.02	0.003	—
	Sev RTT	0	0.001	—
4p‐r	Intact	0.1	0.001	0.05
	Mod RTT	0.032	0.00032	0.0398
	Sev RTT	0	0	0.035
4p‐e	Intact	0.2	0.03	0.05
	Mod RTT	0.1	0.015	0.0375
	Sev RTT	0	0	0.025

Apnoeas in RTT simulations are terminated by noise‐induced activation of the I unit in the 4p‐e model and by noise‐induced deactivation of the KF‐E unit in the 4p‐r model. This distinction in dynamics arises from several key parameter differences between the models. Specifically, compared to the 4p‐e model, (1) the E unit inhibits the I unit more strongly in 4p‐r, providing a stronger suppression of I activation (see parameter b_bc,pbc), (2) the E unit more strongly inhibits the KF‐E in 4p‐r, so that even with some loss of inhibition the KF‐E unit is more susceptible to deactivation by noise (see parameter b _{bc ,kf ‐ e}), and (3) tonic drive to the KF‐E is reduced in 4p‐r, which also enhances the likelihood of KF‐E deactivation (see parameters ρ_{kf ‐ e} and μ_{kf ‐ e}).

Simulations were performed using an adaptive Runge‐Kutta algorithm in the freely available XPPAUT software package (Ermentrout, 2002), sometimes with a MATLAB (versions 9.09.5) interface for iterations over parameter values. All simulations were performed on a typical PC laptop (HP ProBook 450 G1). Most simulations were completed in under an hour. Simulations for Figs 5 and 9 were completed overnight.

Figure 5. Reduction in inhibition to KF‐E simulates Rett syndrome.

Reduction in inhibition to the KF‐E unit leads to increased duration and variability of the respiratory cycle period due to increased expiratory phase durations and variability associated with prolonged KF‐E activation. A–C, period as a function of fraction of inhibition to the KF‐E that is present, relative to the baseline model, in the intact case. The blue curves denote averages while the cyan clouds represent the standard deviations. A, 3p model. B, 4p‐r model. C, 4p‐e model. D, example voltage time courses for the 4p‐e model for the inhibition levels marked with a green circle, yellow square, and red triangle, respectively, in C. Colour keys as in Fig. 4. E–G, periods as a function of fraction of inhibition to the KF‐E that is present, relative to the baseline model, in the vagotomized case.

Figure 9. Breathing periods and standard deviations across all simulated regimes.

Top: 3p model. Middle: 4p‐r model. Bottom: 4p‐e model. Navy blue bars: inspiratory phase duration (T _I). Aquamarine bars: expiratory phase duration (T _E). Yellow bars: total period (T _T). Plus symbols denote simulated application of 5‐HT_1A agonist. For parameters used in the RTT cases shown here, see Table 9. [Color figure can be viewed at wileyonlinelibrary.com]

Phase plane analysis

Output patterns were analysed by projecting network solutions, or trajectories, to phase planes defined by the voltage and persistent sodium inactivation variables for individual neurons and visualizing these together with relevant nullclines. For a system of two differential equations, in the phase plane formed by plotting one dependent variable against the other, each variable's nullcline is the collection of points for which the righthand side of its differential equation equals 0; typically these form curves. Nullclines can help to explain the dynamic mechanisms underlying an observed output pattern. In our models, over at least some parameter range, each V‐nullcline consists of a cubic‐shaped curve in the (V, h) phase plane, with an attracting branch at low voltages corresponding to an inactive or silent phase, an unstable middle branch, and an attracting branch at elevated voltages corresponding to an active phase; if V represents average voltage across a neuronal population, then individual neurons within the population would be spiking in the active phase. When projected to the phase plane of each neuron, a trajectory mostly stays near the lower (silent) or upper (active) branch of each neuron's V‐nullcline, with occasional rapid switches between branches. The concepts of escape and release are useful for considering how these switches can occur (Wang & Rinzel, 1992; Skinner et al. 1994; Daun et al. 2009).

The simple example shown in Fig. 2 illustrates the concepts of escape and release in a network of two neurons coupled by mutual synaptic inhibition. The neurons in the example network are mathematically similar to the model pre‐BötC and BötC neurons, but their parameters are tuned differently (and are unequal to each other) for illustrative purposes. The voltage time courses for the coupled network exhibit transitions in which the two neurons switch their active and silent roles (Fig. 2 A). Importantly, the position of the V ‐nullcline for a neuron in a synaptically coupled network depends on the synaptic inhibition it receives from other neurons (based on the I _syn terms in eqn (1)). Figure 2 B shows example V‐nullclines for cell 1. The voltage nullcline corresponding to full inhibition from cell 2 to cell 1 (maximal s _{2 , 1}) is plotted in continuous red, while the voltage nullcline corresponding to no inhibition to cell 1 (i.e. when cell 2 is inactive) is plotted in the dashed red curve. The cyan curve shows the nullcline for the inactivation variable h for cell 1, based on eqn set (3). Fixed points of the system, for a fixed inhibition level, occur at the intersection of the voltage nullcline corresponding to that inhibition level and the inactivation nullcline. The black cycle is the trajectory from Fig. 2 A projected to the (V ₁ ,h ₁) phase plane. The vertical dotted black line shows the synaptic threshold for synaptic inhibition from cell 1 to cell 2 and from cell 2 to cell 1 (i.e. in this parameter tuning, θ_{syn1 , 2} = θ_syn2,1; see eqn set (4)) When cell 1 is inhibited by cell 2, the projected trajectory evolves slowly up the left branch of the full‐inhibition V ₁‐nullcline. At the point marked with a black square, V ₁ reaches the left local extremum, or knee, of its nullcline, where the silent branch ends. The trajectory then undergoes a rapid excursion to the right, in the direction of increasing voltage, crossing the threshold for synaptically inhibiting cell 2. This type of transition, driven by the voltage of the silent cell, is called an escape. The inhibitory conductance s _{1 , 2} becomes large and this change adjusts the V‐nullcline for cell 2, such that cell 2 transitions to the silent phase and s _{2 , 1} drops close to 0. Thus, the projected trajectory in the (V ₁ ,h ₁) phase plane approaches the dashed V ₁‐nullcline corresponding to the uncoupled dynamics of cell 1 near V ₁ = −10 mV. The trajectory evolves slowly down the right branch of this no‐inhibition V ₁‐nullcline in the (V ₁ ,h ₁) phase plane. At the point marked with a grey circle, it reaches the right knee of the V ₁‐nullcline, where the active nullcline branch ends in another nullcline knee. The trajectory then undergoes another rapid transition, in the direction of decreasing voltage, which brings it through the synaptic threshold and causes s _{1 , 2} to approach 0. This transition driven by the active cell, called release, allows cell 2 to activate.

Figure 2. An example simulation involving transitions by escape and release.

A, voltage time courses for two units coupled by mutual synaptic inhibition. The black square denotes a time when unit 1 activates, while the grey circle marks a time when unit 2 activates. B, projection to the phase plane for unit 1. C, projection to the phase plane for unit 2. In both phase planes, in addition to the model trajectory (grey in B, blue in C), two V‐nullclines are included, one corresponding to the maximal level of inhibition to the unit (continuous red line) and one corresponding to no inhibition to the unit (dashed red line), along with an h‐nullcline (cyan). The threshold for synaptic inhibition is also shown (black dashed line). See Methods for details. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 2 C displays the nullclines and projected trajectory for cell 2. Analogously to cell 1, between rapid jumps in voltage, the trajectory projected to the phase plane for cell 2 evolves along branches of the V ₂‐nullcline, corresponding to the level of inhibition it is receiving. When cell 2 is active, however, it cannot reach the right knee of the uninhibited V ₂‐nullcline (indeed, parameters are tuned such that this nullcline is monotonic, without knees; see dashed nullcline in Fig. 2 C). Thus, cell 2 cannot initiate a transition by release. Similarly, when cell 2 is silent, it cannot reach the left knee of the inhibited V ₂‐nullcline due to the presence of a fixed point, where the V ₂‐ and h ₂‐nullclines intersect (near the grey circle in Fig. 2 C). Thus, cell 2 cannot initiate a transition by escape. In the projection of the trajectory from Fig. 2 A, which is shown in blue, cell 2 jumps to high voltage and thus becomes active only when it is released by cell 1 (grey circle) and jumps down to low voltage and thus becomes inactive only when cell 1 escapes (black square), as described above.

Results

The purpose of this study was to use computational modelling to test the hypotheses that (1) the variable, apnoeic breathing patterns observed in Rett syndrome can arise from loss of inhibition to respiratory neurons in the Kölliker‐Fuse nucleus (KF‐E unit in the model), and (2) the restoration of eupnoea‐like breathing patterns by serotonergic (5‐HT_1A) agonist application in Rett syndrome can arise from these agonists’ influence on 5‐HT‐activated potassium channels (see Methods). To achieve these aims, we utilized minimalistic reduced computational models that allow for clear elucidation of mechanisms underlying simulation outcomes and hence direct mapping of results to mechanistic biological predictions. In this modelling framework, differential equations for a single neuron or unit lacking spiking currents were used as a proxy for the dynamics of each biological neuronal population, justified by the assumption that each relevant population exhibits relatively synchronized transitions between low and high activity states, but with asynchronous spiking that would result in a net averaging out of spikes over the population (see Methods).

Two different base models were developed: one model comprised three potentially rhythmic respiratory neuron populations, including KF‐E, and hence was dubbed the 3p model (Fig. 1 A); the other model supplemented these three populations with a second pontine population from outside of the KF, which we denote with the general label of parabrachial (PB‐I), and thus was called the 4p model (Fig. 1 B). Two 4p model variants (4p‐e and 4p‐r) were considered, distinguished by their parameter tunings (see Methods) but not by their structural features. In each model, the dynamics of the inspiratory pre‐BötC unit and the expiratory Bötzinger complex (BötC) unit depended on a persistent sodium current that allowed rhythmicity of that unit for certain fixed input levels (Smith et al. 2007; Anderson et al. 2016), although similar dynamics would arise from other currents that enhance excitability in the silent phase and feature slow negative feedback (Izhikevich, 2007). Each of the 3p, 4p‐e and 4p‐r models was tuned to produce experimentally observed patterns under several conditions, as described below, and then tested under simulated Rett syndrome, with and without 5‐HT_1A agonist application. Differences between the 4p‐e and 4p‐r models have minimal effects on eupnoeic and vagotomized dynamics and will be discussed in the context of RTT, since simulated RTT unmasks these differences.

Intact models produce eupnoeic breathing patterns

Each of the three models, given its baseline parameter tuning and arbitrary initial conditions, settled into an activity pattern corresponding to a eupnoeic respiratory rhythm (Fig. 3 A and D). Although a specific baseline parameter set was chosen for each model, this pattern was robust to reasonable parameter variations. In all cases, this rhythm consisted of a periodic alternation (at frequencies appropriate for mouse or rat) of phases of elevated inspiratory activity with phases of elevated expiratory activity, such that the expiratory activity duration was about twice the inspiratory activity duration (3p: T _i = 328 ms, T _e = 514 ms; 4p‐r: T _i = 355 ms, T _e = 673 ms; 4p‐e: T _i = 365 ms, T _e = 601 ms). KF‐E activity was suppressed in these patterns, while the additional PB‐I unit in the 4p cases exhibited moderate activity that was approximately tonic but with small surges that did not contribute to rhythm generation, which occurred in response to its inputs from the inspiratory pre‐BötC unit (Fig. 1 B).

Figure 3. Voltage time courses and phase plane views of intact model outputs in the eupnoeic state.

A–C, 3p model. D–F, 4p‐r model (4p‐e model results are almost identical and are not shown). A and D, eupnoeic output patterns feature a longer BötC expiratory activation period than pre‐BötC inspiratory duration as well as pontine suppression (colour keys below panels). The black squares indicate the start of inspiration, the grey circles the start of expiration, in an example cycle. B and E, projections to the phase plane for the pre‐BötC variables. In each, the trajectory (grey), three V‐nullclines (red), and one h‐nullcline (cyan) are shown. The continuous V‐nullcline corresponds to the maximal level of inhibition to the pre‐BötC unit, received at the start of its silent phase (i.e. expiration). The dashed V‐nullcline applies in the absence of inhibition, corresponding to its active phase (i.e. inspiration). The dashed‐dotted V‐nullcline corresponds to the level of inhibition received when the pre‐BötC unit starts to transition to the active phase (at the moment indicated by the black square). The vertical dashed black line indicates the synaptic threshold ${\theta }_{{\mathrm{syn}}_{\mathrm{pbc},\mathrm{bc}}}$ (see Methods). The dashed‐dotted nullclines are almost indistinguishable from the continuous ones, except in the insets. C and F, projections to the phase plane for the BötC variables. Nullcline coding is the same except that the continuous V‐nullcline now corresponds to the start of inspiration, the dashed to the start of expiration, and the dashed‐dotted to the start of the transition to expiration (at the moment indicated by the grey circle), such that the colour key below is correct for panels B, C, E, and F. The vertical dashed black line indicates the synaptic threshold ${\theta }_{{\mathrm{syn}}_{\mathrm{bc},\mathrm{pbc}}}$.

We used nullclines to understand the dynamic mechanisms underlying the models’ eupnoeic rhythmicity, which in our simple modelling framework depended on the persistent sodium current, I _NaP. Since the I _NaP inactivation variable h evolves much more slowly than voltage, the voltages of the pre‐BötC and BötC units would rapidly equilibrate to quasi‐steady states, with subsequent voltage dynamics slaved to h except during rapid switches between inspiration and expiration. Thus, the model trajectory projected to the phase space variables of one unit, such as (V _pbc ,h _pbc) for the pre‐BötC unit or (V _bc ,h _bc) for the BötC unit, would generally lie on the voltage nullcline, or curve of zero voltage time derivative, of that unit except during the fast switches. Which nullcline was selected at any time corresponded to the level of inputs received by the unit at that time (see Methods).

In the intact 3p model (Fig. 3 A–C), although the BötC unit could oscillate in the absence of inhibition, it received enough inhibition during inspiration to prevent it from activating (grey circle, Fig. 3 C). The switch from inspiration to expiration occurred when the voltage of the active pre‐BötC unit hyperpolarized to a level close to the synaptic threshold (black dashed line, Fig. 3 B), corresponding to adaptation after an extended active period (grey circle, Fig. 3 B). The resulting loss of inhibition allowed the voltage nullcline of the BötC unit to settle to a lower position (dashed‐dotted red curve, Fig. 3 C), such that the unit could transition to the active phase. After this activation, the BötC unit inhibited the KF‐E unit, causing KF‐E to remain inactive, and provided just enough inhibition to prevent the activation of the pre‐BötC unit (black square, Fig. 3 B). The switch from expiration to inspiration thus relied upon a small degree of adaptation of the active BötC unit, which lowered the voltage nullcline of the pre‐BötC unit enough to allow it to activate (Fig. 3 B, inset), although much less adaptation was needed than during the switch from inspiration to expiration. Finally, during inspiration, feedback signals through the vagal pathway via the NTS inhibited KF‐E, maintaining its inactivity and its lack of contribution to the eupnoeic rhythm.

The 4p‐e and 4p‐r models produced almost identical eupnoeic rhythms to each other. In these activity patterns, the switch from expiration to inspiration was similar to that in the 3p model (black squares, Fig. 3 E and F; inset, Fig. 3 E), as was the suppression of the KF‐E unit during both expiration and inspiration. Unlike the 3p case, however, the switch from inspiration to expiration occurred without adaptation when the suppressed BötC unit, despite being fully inhibited by the active pre‐BötC unit, was able to reach the fold of its voltage nullcline and enter the active phase by escape (grey circles, Fig. 3 E and F; see Methods for discussion of transitions by escape); that is, although the BötC unit was not intrinsically rhythmic without input, inhibition yielded sufficient deinactivation of persistent sodium to allow it to eventually activate, highlighting a possible role for slow inward currents even in neurons lacking intrinsic rhythmicity (related to post‐inhibitory rebound, cf. Getting, 1989; Daun et al. 2009). The tuning of parameters to allow this escape was necessary for the model to produce both eupnoeic outputs with KF‐E suppression under baseline conditions and appropriate rhythmic outputs under vagotomy despite the loss of vagal input, without requiring an intrinsically rhythmic KF population (see next section).

Models produce slower breathing patterns with prolonged expiration under simulated vagotomy

Various experimental models study respiratory rhythm generation after vagus nerve transection. Despite the loss of vagal input to respiratory neural populations, this manipulation yields respiratory rhythms maintaining the basic inspiratory‐expiratory phase alternation, but with a longer period and with increased ratio of expiratory duration to inspiratory duration (Monteau et al. 1990; Connelly et al. 1992; Dick et al. 2008). Our model networks all reproduced these features (Fig. 4 A and E). In all cases, the KF‐E unit in the model became rhythmic, activating during expiration and falling inactive during inspiration (green traces, Fig. 4 A and E) In the 4p models, KF‐E activity alternated with PB‐I activity, such that the PB‐I unit was active during inspiration and inactive during expiration (hence the labels E for the KF‐E unit and I for the PB‐I unit). Different dynamic mechanisms gave rise to the vagotomized rhythms in the 3p and 4p models, however.

Figure 4. Voltage time courses and phase plane views of vagotomized model outputs in the eupnoeic state.

Colour keys and labels as in Fig. 3. A–D, 3p model. E–H, 4p‐r model (4p‐e model results are almost identical and are not shown). A and E, eupnoeic outputs under vagotomy feature a longer expiratory phase and ratio of expiratory duration to inspiratory duration than in the intact case (Fig. 3), with rhythmic KF‐E and PB‐I alternation that is absent in the intact case. Insets show that in the 3p model, KF‐E activation occurs at the start of expiration (A) while in the 4p models, it follows BötC activation. B and F, projections to the pre‐BötC phase plane. C and G, projections to the BötC phase plane. D and H, projections to the KF‐E phase plane. The continuous V‐nullcline corresponds to the level of inhibition received during inspiration, the dashed V‐nullcline to the level received during expiration.

In the 3p case (Fig. 4 A–D), the loss of vagal input rendered the BötC unit unable to activate even after the adaptation of pre‐BötC activity (grey circles, Fig. 4 B and C). In the absence of vagal inhibitory feedback signals, however, the KF‐E unit was able to transition autonomously into the active phase (grey circle, Fig. 4 D). Once this activation occurred, the excitation from the KF‐E recruited BötC activity and helped sustain a prolonged expiratory phase. The inhibition from the BötC unit to the KF‐E (Fig. 1) did not interfere with this maintained expiration, since the voltage nullcline of the KF‐E unit, even in the presence of this inhibition, featured an extended active phase branch (Fig. 4 D, dashed V‐nullcline). Once the KF‐E unit finally reached the end of this branch, it became inactive again, removing a source of excitation to the BötC unit and causing the BötC unit to become inactive, thereby releasing the pre‐BötC unit and allowing inspiration to commence (black squares, Fig. 4 B–D).

The 4p models were specifically designed to explore the alternative hypothesis that the KF‐E population, even in the absence of vagal feedback inhibition, could not autonomously generate rhythmicity. This constraint is manifested in the existence of a stable fixed point on the left branch of the KF‐E unit's voltage nullcline in the absence of input in the 4p case (grey circle, Fig. 4 H). In the 4p models, parameters were tuned such that in the absence of vagal feedback excitation, the BötC unit was still able to transition to the active phase once the pre‐BötC unit adapted sufficiently (grey circles, Fig. 4 F and G; see also inset of Fig. 4 E, which illustrates that BötC unit activation precedes KF‐E activation) Note that the BötC activation here does require that it receives a sufficiently strong tonic drive, some of which could come from pontine sources (Alheid et al. 2004). Without KF‐E, our model would still produce a rhythm but with a greatly shortened E phase (analogous to the 2‐phase rhythm seen experimentally with pontine transection; Rybak et al. 2007; Smith et al. 2007), whereas the loss of all pontine drive to the BötC unit yielded apneusis. The BötC unit's transition to the active phase was delayed relative to the eupnoeic case, resulting in an increased inspiratory duration in vagotomized rhythms relative to eupnoeic rhythms, which stands as a prediction of the 4p models. In this case, the loss of pre‐BötC activity removed the drive to the PB‐I unit and hence its inhibition of the KF‐E unit as well. This loss of inhibition lowered the KF‐E unit's voltage nullcline relative to its position with inhibition from the PB‐I and BötC present (Fig. 4 H, dashed V‐nullcline) and subsequently allowed the KF‐E unit to activate fully, despite the inhibition it received from the BötC unit. Once it was active, the excitation from the KF‐E to the BötC pinned the BötC in the active phase and correspondingly prolonged expiration, and enhanced the adaptation of the BötC unit, until the KF‐E activity adapted and the KF‐E unit became inactive again (black square, Fig. 4 H). Finally, the loss of excitation from KF‐E to BötC, following the inactivation of KF‐E, caused the BötC unit to become inactive (Fig. 4 G) and released the pre‐BötC unit to activate and initiate the next phase of inspiration (Fig. 4 F), as in the 3p model. Thus, this model does include a contribution of KF‐E to the transition from expiration to inspiration.

Several predictions come out of the differences between the models, their dynamic mechanisms, and their parameter tunings in the vagotomized regime. First, in the 3p case, blockade of KF‐E activity should prevent the termination of inspiration, whereas in the 4p cases, expiratory interruptions of inspiration would still be possible without KF‐E participation, although they might be brief. Correspondingly, a surge in KF‐E activity would be predicted to precede activation of BötC in the 3p but not the 4p cases (Fig. 4 A and E: compare insets). Moreover, the inspiratory off‐switch would be more robust to the application of hyperpolarizing current to the BötC in the 3p case than in the 4p cases. Second, as we have already noted, the 4p models predict that vagotomization will increase the duration of inspiration, whereas the 3p model does not. Figure 9 provides a comparison of inspiratory, expiratory, and total cycle durations across models and conditions and clearly makes the comparison of the intact and vagotomized cases (additional scenarios shown in the figure will be discussed below). Third, the 3p model predicts, somewhat counterintuitively, that under vagotomy the KF‐E is subject to stronger inhibitory inputs during expiration than during inspiration, whereas the 4p models predict the opposite (Fig. 4 D and H: compare relative positions of continuous and dashed V‐nullclines). Fourth, due to the differing influences of the KF‐E unit on the voltage nullcline of the BötC unit while both are active, the 3p model predicts that expiration can continue slightly beyond the inactivation of the KF‐E, whereas any such extension would be absent or extremely limited in the 4p models.

Models produce activity patterns characteristic of RTT under block of inhibition to the KF‐E

Experimental studies have suggested that a loss of GABAergic inhibition to respiratory neurons within the KF‐E may represent a key factor in the emergence of the altered breathing patterns associated with RTT and the Mecp2 knockout mouse model (Medrihan et al. 2008; Abdala et al. 2010, 2016). Our computational models are designed to test the viability of this idea. To do so, we gradually reduced the strength of various inhibitory connections targeting the KF‐E unit in the model (Fig. 1, connections marked with R; see Table 9) and observed the resulting model dynamics. In all cases, we reduced the strength of the direct inhibitory projection from the BötC unit to the KF‐E. In our simulations of RTT conditions within an intact system, we also reduced the strength of the connection representing inhibition from the NTS to the KF‐E; this connection was absent in our simulations of RTT conditions in a vagotomized state. Finally, in the 4p models, we also reduced the intrapontine inhibition from the PB‐I to the KF‐E, although this inhibition was not reduced all the way to zero, reflecting the likely contribution of glycinergic inhibition within the pons. The strength of this inhibitory connection was reduced by up to 50% in the 4p‐e model and 30% in the 4p‐r model. With more severe reductions, the BötC neurons in both 4p models lost the ability to escape in the vagotomized regime without excitatory drive from the KF‐E. As the 4p models were designed to explore the hypothesis that the BötC neuron is able to escape without the KF‐E, we selected the reductions accordingly, informed by knowledge of the presence of multiple inhibitory transmitters in the pons (Morschel & Dutschmann, 2009; Dutschmann & Dick, 2012). In all cases in which multiple connections were altered, parameter changes were made proportionately using a scaling factor; if we let α ∈ [0,1] denote the fraction of total inhibition present, then each affected connection strength was set to p _min + α(p _max − p _min), where p _min and p _max denote the minimum and maximum values, respectively, for that strength. All three models exhibited an increase in expiratory duration and variability as inhibition was reduced, as detailed below; furthermore, consistent with experiments (Stettner et al. 2007; Abdala et al. 2016), glutamate injection to the KF‐E, simulated by turning on a tonic excitatory current with reversal potential 0 mV, prolonged expiration, with a stronger effect in RTT than in control conditions, in all of the models (data not shown).

The 3p and 4p‐r models in the intact case showed qualitatively similar behaviours as inhibition was reduced (Fig. 5 A and B). After an initial interval of insensitivity to changes in inhibition, an interval of inhibition levels occurred in which usual cycles were occasionally interrupted by apneic cycles (i.e. cycles with prolonged expiration), leading to an increase in average period and in variability. Below this inhibition interval (with approximately 20–50% of inhibition remaining for 3p and 10–30% of inhibition intact for 4p‐r), regular rhythmicity re‐emerged, but with prolonged expiratory duration. Finally, for inhibition below these levels, apnoeas became even longer and more irregular, with progressively more variability and increased mean apnoea duration as inhibition was progressively decreased. In the 3p case, once inhibition dropped below ∼10% of normal, rhythmicity was lost, whereas in the 4p‐r case, rhythmicity was maintained all the way down to α = 0 (Fig. 5 A and B).

The 4p‐e model also yielded increased variability and apnoea duration, leading to corresponding changes in period, as inhibition to the KF‐E was gradually diminished, with a qualitatively similar progression to the 4p‐r and 3p cases (Fig. 5 C). The three abnormal regimes in the 4p‐e model are illustrated in Fig. 5 D (green circle: occasional apnoeas; yellow square: regular apnoeas, called mod Rett in Table 9 and in Fig. 9; red triangle: variable duration, generally prolonged apnoeas, called sev Rett in Table 9 and in Fig. 9). The 4p‐e case featured more of a gradual transition from the regime in which all expiratory cycles were prolonged (yellow square) to the highly variable regime (red triangle) than the other models, in that it featured some variability in period throughout the entire corresponding range of inhibition levels.

The activity patterns of the 3p and 4p‐r models were less similar under our simulations of the gradual loss of inhibition to the KF‐E in the vagotomized case. In the 3p model, because the KF‐E activated to initiate expiration and only became inhibited after expiration was underway, the loss of up to 80% of inhibition had almost no effect on network outputs (Fig. 5 E). From inhibition levels at 20% of baseline down to 10% of baseline where rhythmicity was lost, significant apnoeas with extensive variability could finally emerge, due to prolonged KF‐E activation. Like the 3p model, the vagotomized 4p‐r model also showed a loss of the intermediate regime of variable expiratory durations after initial apnoea onset (compare Fig. 5 A and B versus E and F). In the 4p‐r case, however, variability appeared with a much larger fraction of inhibition still remaining and became progressively more extreme, with corresponding increases in mean respiratory cycle period. Finally, the vagtomized 4p‐e model did not show much variability until inhibition had dropped by about 50%, after which an abrupt increase in variability and average period occurred, which was maintained down to complete inhibition blockade (Fig. 5 G). Once the initial onset of variability had arisen, the progression for the vagtomized 4p‐e model was quite similar to that for the intact 4p‐e model (compare Fig. 5 C vs. G).

We can partially explain the differences between the 4p‐r and 4p‐e models by consideration of model activity patterns in appropriate phase planes. In our simulations of severe RTT (i.e. sufficient reduction of inhibition to KF‐E), the lessened inhibition of the KF‐E yields a stable fixed point, corresponding to tonic activation of KF‐E (black circle, Fig. 6) and suppression or inactivity of the pre‐BötC unit (yellow circle, Fig. 7). Because noise is included in the system, projections of trajectories to these phase planes do not approach the fixed point asymptotically but rather approach a small neighbourhood of this fixed point. Orbits in this neighbourhood exhibit small‐amplitude oscillations because the fixed point is a stable spiral point. Noise produces an effective voltage threshold, such that if, on a particular oscillation cycle, the projection of the orbit to the KF‐E phase plane drops below the KF‐E voltage associated with this threshold, then KF‐E activation terminates, inducing a corresponding loss of BötC activation. On the other hand, if the projection of the orbit to the pre‐BötC phase plane crosses above the pre‐BötC voltage associated with the threshold, then the pre‐BötC unit can activate.

Figure 6. Simulation of severe RTT results in a stable equilibrium point with sustained KF‐E activation.

In the projection to the KF‐E phase plane, the V‐nullcline that is relevant during expiration (continuous red line) intersects the h‐nullcline (cyan) on the right branch of the V‐nullcline, yielding a stable equilibrium point (black circle). The figure also includes a projected cycle of an apnoeic solution (green) and the V‐nullcline corresponding to the application of a small, depolarizing current to the KF‐E unit (dashed‐dotted red). Inset: a zoomed view near the equilibrium point reveals that the projection of the model output winds around this equilibrium due to noise (arrows show direction of net rotation). On each cycle, if fluctuations pull voltage low enough (far enough to the left), then a transition out of expiration will result (from region marked ‘EXIT’). Otherwise, another cycle must occur before this transition can be possible. Successive expiratory phases, each consisting of multiple cycles, are coloured black, cyan and red.

Figure 7. Simulation of severe RTT results in a stable equilibrium point with sustained pre‐BötC suppression.

In the projection to the pre‐BötC phase plane, the V‐nullcline that is relevant at the start of expiration (dashed red line) intersects the h‐nullcline (cyan) on the left branch of the V‐nullcline, yielding a stable equilibrium point (yellow circle). The figure also includes a projected cycle of an apnoeic solution (grey), the V‐nullcline relevant during inspiration (continuous red), and the V‐nullcline corresponding to the application of a small, hyperpolarizing current to the pre‐BötC unit (dashed‐dotted red). Inset: a zoomed view near the equilibrium point reveals that the projection of the model output winds locally due to noise (arrows show direction of net rotation); furthermore, the V‐nullcline location and hence the equilibrium location as well drift to larger V and smaller h due to gradual adaption of BötC activity and decrease of inhibition. On each cycle, if fluctuations pull voltage high enough (far enough to the right), then a transition into inspiration will result (from region marked ‘EXIT’). Otherwise, another cycle must occur before this transition can be possible. Successive expiratory phases, each consisting of multiple cycles, are coloured black, cyan and red.

In the 4p‐r model, parameters are tuned such that the KF‐E active phase right branch fixed point lies near the right knee of its V ‐nullcline in the active phase, which strongly favours the former scenario of expiratory phase termination by KF‐E and BötC deactivation (see Methods for full details of parameter differences between 4p‐r and 4p‐e models). This deactivation releases the pre‐BötC unit from inhibition and thus allows it to activate, hence the name 4p‐r, which refers to 4p with release. In the intact case, a significant reduction of inhibition to KF‐E is needed before the stable fixed point and associated prolonged, variable expiratory phase duration can arise, as seen in Fig. 5 B. In the vagotomized case, noise can prolong KF‐E activation even for relatively large inhibition levels, for which the fixed point is not stable but is close to a bifurcation that will stabilize it. As inhibition gradually decreases, the KF‐E activation level becomes more elevated relative to the threshold for KF‐E deactivation, such that deactivation becomes less likely and the magnitude and variability of apnoea duration increase (Fig. 5 F).

In the 4p‐e model, parameters are tuned such that the fixed point for the pre‐BötC unit under full inhibition lies near the left knee of its V‐nullcline, which strongly favours noise‐induced activation, or escape, of the pre‐BötC unit, hence the name 4p‐e for 4p with escape (see Methods). This effect can arise fairly similarly in the intact and vagotomized cases. In the intact case, however, alterations to the basic rhythm can emerge with less reduction of inhibition to the KF‐E because KF‐E activation starts to occur on occasional cycles (Fig. 5 D, green circle); in the vagotomized case, KF‐E already activates on each cycle, so there is no opportunity for this transitional regime. Note that in the 4p‐e model, during an extended apnoea, some adaptation of the BötC unit occurs, yielding a corresponding mild reduction in inhibition to the pre‐BötC unit. This change allows the fixed point for the pre‐BötC unit to drift to slightly lower h _pre‐I values, with a corresponding drift in the vertex of the spiralling trajectory observed in projection to the pre‐BötC phase plane (Fig. 7) and in the effective voltage threshold for pre‐BötC unit activation.

Finally, the mechanistic difference in the apnoea termination mechanisms in the 4p‐r and 4p‐e models gives rise to some clear differences in the predictions they make about how the respiratory neural network will respond to perturbations in RTT conditions. In the 4p‐r model, the initiating step in apnoea termination is a fluctuation‐induced loss of KF‐E activation (Fig. 6), while in the 4p‐e model, termination originates with the onset of pre‐BötC activation (Fig. 7). Therefore, in the 4p‐r model, the application of weak depolarizing stimulation to the KF‐E unit, by preventing the loss of KF‐E activation, yields sustained KF‐E and BötC activity and prevents the termination of expiration (Fig. 8, upper left). Alternatively, the application of weak hyperpolarizing input to the pre‐BötC unit has little effect on apnoea properties in the 4p‐r model, since this stimulation does not interfere with the loss of KF‐E activation (Fig. 8, upper right). In contrast, in the 4p‐e model, the application of weak depolarizing stimulation to the KF‐E unit does not noticeably affect apnoea termination, since this stimulation does not impact on the ability of the pre‐BötC unit to activate (Fig. 8, lower left). The application of weak hyperpolarizing input to the pre‐BötC in the 4p‐e model, however, will suppress pre‐BötC activation and hence lead to sustained KF‐E and BötC activation and expiration (Fig. 8, lower right).

Figure 8. Stimulation experiments in RTT can distinguish the 4p‐e and 4p‐r models.

Top row: voltage time courses for the 4p‐e model. Bottom row: voltage time courses for the 4p‐r model. Left column: results of applying a small hyperpolarizing current to the pre‐BötC unit. The 4p‐e model predicts that this stimulation should pin the system in sustained apnoea, whereas the 4p‐r model predicts that this modulation should have little effect. Right column: results of applying a small depolarizing current to the KF‐E unit. The 4p‐e model predicts that this stimulation should have little effect, whereas the 4p‐r model predicts that this modulation should pin the system in sustained apnoea. Colour key below applies to all panels. [Color figure can be viewed at wileyonlinelibrary.com]

Application of a 5‐HT_1A agonist in simulated RTT normalizes expiration

Our simulation results show that sufficient reduction of inhibition to the KF‐E in simple respiratory neural models induces the prolonged expiration and increase in expiratory variability associated with breathing in RTT. Experimental work has suggested that loss of 5‐HT_1A‐mediated inhibition could induce respiratory cycle irregularity (Dhingra et al. 2016), that application of 5‐HT_1A agonists improves cycle regularity even in wild‐type mice (Stettner et al. 2007), and that such agonist application could alleviate variability and prolonged expiration in RTT (Levitt et al. 2013; Abdala et al. 2014 a, b ). Hence, we next explored the effects of simulated 5‐HT_1A agonist application in our models. To represent this intervention, we strengthened inhibitory connections within the medullary kernel as well as the glycinergic component of the inhibition from the PB‐I to the KF‐E that was maintained in our simulated RTT conditions and, following previous computational modelling (Shevtsova et al. 2011), we activated a 5‐HT_1A‐sensitive hyperpolarizing (e.g. potassium) current in all populations. Although the adjusted inhibition levels modified V‐nullcline positions, the dominant effect of this modification occurred via the potassium channel adjustment. This change caused a significantly earlier, stronger adaptation of the BötC unit during expiration, shortening expiratory durations to near the levels associated with intact eupnoeic breathing in the moderate RTT case and towards the levels associated with vagotomized eupnoeic breathing in the severe RTT case (Fig. 9). For comparison, we also simulated 5‐HT_1A agonist application in the intact and vagotomized eupnoeic cases. Interestingly, in the 4p models, we observed a slight lengthening of inspiration in the intact case and a slight shortening of expiration in the vagotomized case, which reflects a stronger effect of 5‐HT_1A‐sensitive potassium channels on the BötC unit than on the pre‐BötC unit and stands as a prediction of our model tuning (Fig. 9).

Discussion

RTT leads to dysfunctional breathing that includes extended, variable apnoeas. Identifying the source of these breathing disruptions would be a key step in developing therapeutic interventions and would also provide insights about basic properties of respiratory rhythm generation and control. This study takes a novel step in the investigation of RTT: we have harnessed a reduced computational modelling framework, which allows direct manipulation of specific network components without confounds associated with experimental techniques, to perform simulations spanning multiple network configuration states and a clinical intervention. Our results demonstrate that loss of GABAergic inhibition to the KF nucleus could underlie the prolongation and variability of expiration in RTT, as suggested by recent experiments (Abdala et al. 2010, 2016), and that effects of 5‐HT_1A agonists on inhibitory interactions and membrane potentials in the respiratory neural circuit suffice to alleviate this respiratory pathology (Levitt et al. 2013; Abdala et al. 2014 a, b ). We obtained these results using not just one but three related but distinct models, each of which makes distinct predictions about which key pathways and interactions allow these effects to emerge (Fig. 1), providing natural targets for exploration in future experiments. Our findings also make predictions that go beyond scenarios considered in previous experiments, specifically suggesting that respiratory outputs should show some resilience to the gradual reduction of inhibition to the KF, followed by a gradual enhancement of expiratory duration and variability (Fig. 5), all of which can be tested in future experiments to evaluate how the loss of GABAergic inhibition to the KF contributes to the respiratory alterations observed in RTT.

Although our models are simplified, they are well grounded in experimental results and past literature. The medullary rhythm generation circuit within the model relies on mutually inhibitory synaptic connections between inspiratory and expiratory units. There is a long history of including such interactions in conceptual and computational models of respiratory neural circuitry (reviewed by Lindsey et al. 2012), with recent new insights underlining their importance (e.g. Molkov et al. 2014; Marchenko et al. 2016; Harris et al. 2017; Ausborn et al. 2018; Baertsch et al. 2018; Ramirez & Baertsch, 2018). An earlier large‐scale model that reproduced a wide range of experimental results incorporated a projection from medullary expiratory units to the pons (Rybak et al. 2008) as is present in our model, while additional medullary‐pontine and intra‐pontine interactions that we included are also based on previous work combining experiments and theory (Dick et al. 2008; Morschel & Dutschmann, 2009; Dutschmann & Dick, 2012; Barnett et al. 2018). Our model incorporates a simplified representation of feedback pathways through the NTS that are known to provide inhibitory inputs to the pons (Ezure & Tanaka, 2004; Kubin et al. 2006), with additional control provided by sources of tonic drive in the medulla and pons (Alheid et al. 2004; Smith et al. 2007). The reduced modelling framework that we utilized has been shown in several previous studies of respiratory and locomotor circuits to match, qualitatively and in some cases quantitatively, the activity of large‐scale, detailed computational model networks and to reproduce experimentally observed features of rhythmic outputs (Rubin et al. 2009, 2011; Molkov et al. 2015; Bacak et al. 2016; Danner et al. 2017; Ausborn et al. 2018). The model in this work, despite its simplicity, produces the respiratory patterns associated with eupnoea and vagotomy and also reproduces the finding that glutamate injection to the KF can prolong expiration, with a stronger effect in RTT than in control conditions (Abdala et al. 2016). In all of the models developed in this work, once inhibition to the KF is decreased sufficiently, the expiratory period increases rather abruptly, leading first to slowed, regular breathing followed, with additional loss of inhibition, by progressively slower and more variable respiration. The dependence of respiratory pattern on inhibition level to the KF points to a possible factor that could contribute to the variability of respiratory phenotypes across individuals with RTT (Julu et al. 2008) and could also have implications for intra‐individual variabilities of the respiratory phenotype, including differences in breathing during sleep compared to wakefulness (Southall et al. 1988; Weese‐Mayer et al. 2008). Since our model omits blood gas exchange and detailed feedback pathways, however, it cannot capture shallow breathing effects.

The role of the KF in the inspiration‐expiration switch under vagotomized conditions represented a key distinction between the 3p and 4p models that we considered. There is evidence suggesting that some neurons in the KF start firing at the end of inspiration (Song et al. 2015), consistent with the idea that some degree of intrinsic rhythmicity may be present within the pons (Poon & Song, 2014). Our 3p model incorporates this viewpoint (Fig. 4 A–D). This contention is a matter of debate, however, and many models do not incorporate intrinsic pontine rhythmicity (e.g. Molkov et al. 2017; Barnett et al. 2018; Ramirez & Baertsch, 2018). Our 4p models illustrate that even in a highly reduced formulation, such rhythmicity need not be present to achieve appropriately timed rhythms under control and vagotomized conditions; in these models, the expiratory BötC unit activates before the KF, while KF activity significantly prolongs expiration (Fig. 4 E–H). As a result of these structural differences, several predictions emerge that could be used to test the relative validity of these models. The 3p model predicts that blockade of KF activity would entirely prevent termination of inspiration in vagotomized preparations, whereas a brief expiratory output would still occur in the 4p models. In the 3p model, expiration could also continue slightly beyond the termination of KF activity, since this termination could arise through mechanisms local to the pons. Finally, the 4p models, but not the 3p model, predicts that inspiratory duration should become prolonged along with expiratory duration under vagotomized conditions (Fig. 9). The two 4p models themselves differ in a way that emerges under simulated RTT conditions. In the 4p‐e model, the prolonged apnoeas that result are terminated when the pre‐BötC unit activates by escaping from ongoing inhibition from the BötC. In contrast, in the 4p‐r model, apnoeas end when the KF and BötC reduce their activity, releasing the pre‐BötC unit to initiate inspiration. This fundamental distinction, illustrated in Figs 6 and 7, yields completely opposite predictions about responses to small inputs to the pre‐BötC and BötC units in RTT (Fig. 8) that could be tested experimentally.

Despite the current and past success of this reduced modelling framework, our model clearly has a variety of limitations. To focus on the inclusion of pontine components and phase plane analysis, we omitted fast spiking currents, noting that past works have shown similar rhythmic behaviours from the spiking and non‐spiking frameworks (e.g. Rubin et al. 2009; Molkov et al. 2015; Bacak et al. 2016; Ausborn et al. 2017); future work should consider effects of spiking currents. We also omitted certain features of the respiratory circuit entirely, such as inhibitory neurons within the pre‐BötC that may play a role in modulating its activity (Baertsch et al. 2018; Ramirez & Baertsch, 2018) and the augmenting expiratory population that extends expiration in more quantitatively accurate models (e.g. Rybak et al. 2007; Smith et al. 2007; Lindsey et al. 2012; Molkov et al. 2017). With the latter omission, we felt it was not appropriate to explore recruitment of abdominal outputs and forced expiration such as in hypercapnia, as has been discussed by others (Molkov et al. 2010, 2014; Rubin et al. 2011; Ramirez & Baertsch, 2018) and as has recently been explored in connection with the KF (Barnett et al. 2018). This omission also limits the utility of our current model to study how the durations of inspiration and expiration and the overall respiratory period will change with modulation of tonic drive strengths, as done in previous work (Rubin et al. 2009). For example, in our 4p models, the BötC unit must escape from the inhibitory influence of the pre‐BötC to initiate expiration in order to obtain appropriate rhythmicity in both eupnoeic and vagotomized conditions, and this reliance on escape, rather than active release initiated by the pre‐BötC, could alter frequency responses to changes in inputs. This assumption is, in fact, consistent with the existence of a rhythmic post‐inspiratory complex (Anderson et al. 2016; Ramirez & Baertsch, 2018); note that while our BötC unit included a persistent sodium current, the unit's intrinsic dynamics differed across models and tonic drive levels (see Table 1). Moreover, this assumption does not impact the relevance of our model for our goals of studying effects of reduction of inhibition to the KF and of 5‐HT_1A agonist application.

Another limitation of this work is that while we checked that our major findings are robust to local variations in model parameter values, we did not do a global exploration of parameter space. The quantitative details of how respiratory periods change with loss of inhibition to the KF (Fig. 5) may be characteristic of the tuning that we used for our model, and variations may arise with other tunings that produce reasonable eupnoeic and vagotomized outputs. Thus, experimental testing of these specific details should not be employed to arbitrate between or rule out our models entirely, but rather to evaluate the validity of our models with the particular parameter tunings used in this paper. Furthermore, we do not claim that reduced inhibition to the KF is the only factor in respiratory disruptions associated with RTT. Our model simply provides a principled demonstration that this reduction may suffice to induce the disruptions, and it yields novel predictions about possible changes in activity as this inhibition is gradually reduced.

Indeed, several other possible mechanisms could contribute to experimental observations that were beyond the scope of our reduced models. We did not explicitly model separate GABAergic and glycinergic inhibitory populations and hence did not test possible 5‐HT_1A agonist effects on the open probability of glycinergic channels (Manzke et al. 2010); however, evidence supporting the extrapolation of these effects to the full respiratory network or to the KF is so far lacking (Rojas & Fiedler, 2016). We also did not consider the role of endogenous serotonergic drive in the KF and elsewhere. It is well established that patients with RTT have reduced serotonergic metabolites in their cerebral spinal fluid (Samaco et al. 2009). It is possible that diminished endogenous serotonergic drive itself could contribute to KF overdrive and breathing instability as demonstrated in some experimental settings (Richter et al. 2003; Dhingra et al. 2016). Another significant simplification in our model was our representation of the feedback pathways in the respiratory circuit, the details of which are still being determined experimentally. It would be reasonable to infer that in patients with RTT, chronic exposure to intermittent hypoxia could cause maladaptive changes, independent from MECP2 deficiency, which could contribute to the instability of the rCPG. It is well established that exposure to chronic intermittent hypoxia (such as in obstructive sleep apnoea) promotes carotid body hyperreflexia and hypertonicity (reviewed elsewhere: Iturriaga et al. 2017). Chronic exposure to hypoxia has also been shown to produce serotonin‐dependent plastic potentiation of centrally integrated peripheral chemoreceptor input (Ling et al. 2001). This notion is corroborated by the observation that mouse models of RTT have enhanced ventilatory responses to hypoxia. However, confirmatory human data is lacking (Bissonnette et al. 2014), and the contribution of these mechanisms to RTT breathing pathology is less clear since hyperoxia, which unloads peripheral chemoreceptors, did not improve periodic breathing in MECP2 deficient mice (Bissonnette & Knopp, 2008).

Finally, the current model also omits the possible contribution of glia to the RTT pathology. For instance, it was demonstrated that CO₂‐induced calcium currents were dramatically reduced in MECP2 deficient astrocytes in the ventral medulla (Turovsky et al. 2015). Medullary astrocytes can contribute to RTN‐mediated ventilatory responses to CO₂ in vivo (Guyenet & Bayliss, 2015). Indeed, MECP2‐deficient mice have impaired hypercapnic ventilatory responses and elevated apnoeic thresholds (Toward et al. 2013; Bissonnette et al. 2014). Moreover, the conditional depletion of MECP2 in astrocytes alone recapitulated the blunted CO₂ sensitivity phenotype (Garg et al. 2015). Interestingly, re‐expression of MECP2 either in glia or in GABAergic neurons alone sufficed to ameliorate the breathing phenotype in mouse models of RTT (Lioy et al. 2011; Ure et al. 2016). This result is consistent with the idea that the balance of drives to expiratory versus inspiratory neuron populations is the most important component of the breathing instability in RTT, a feature that is reproduced by our model.

Reduced models have recently served as useful tools in testing theories of respiratory rhythm generation and control. The results that we obtained by harnessing this framework add support for the ideas that a reduction in inhibition to the KF may play a significant role in producing the disordered respiration associated with RTT, while application of 5‐HT_1A agonists in RTT conditions may enhance respiratory function through contributions of 5‐HT_1A‐sensitive potassium currents or other hyperpolarizing effects. Our findings highlight the importance of future experiments to continue to explore these ideas and suggest possible directions for these experiments to pursue.

Additional information

Competing interests

A. Abdala has received contract research funds from Neurolixis, Inc. on unrelated past projects and has provided paid consultancy to Neurolixis Inc. in the past.

Author contributions

S.W. contributed to the study conception, model construction and tuning, generation and interpretation of data, creation of figures, and manuscript drafting and editing. A.A. contributed to the study conception, interpretation of data, and manuscript drafting and editing. J.R. contributed to the study conception, model construction and tuning, interpretation of data, and manuscript drafting and editing. All authors have approved the final version of the manuscript and agree to be accountable for all aspects of the work. All persons designated as authors qualify for authorship, and all those who qualify for authorship are listed.

Funding

This work was partially supported by the US NSF awards DMS 1312508 (J.R., S.W.) and 1612913 (J.R.) and by the National Centre for Complementary and Integrative Health (NCCIH), NIH award R01AT008632‐01 (A.A.).

Biography

Sam Wittman graduated from the University of Pittsburgh in 2017 with BS degrees in mathematical biology and neuroscience. As an undergraduate, he performed modelling research on the neural basis of Rett syndrome under Dr Jonathan Rubin, and he worked in the vestibular electrophysiology lab of Dr Bill Yates and Dr Andrew McCall. He is currently pursuing a MS in Biostatistics at the Pitt School of Public Health.