A mechanistic model of calcium homeostasis leading to occurrence and propagation of secondary brain injury

Abstract

Secondary brain injury (SBI) refers to new or worsening brain insult after primary brain injury (PBI). Neurophysiological experiments show that calcium (Ca²⁺) is one of the major culprits that contribute to neuronal damage and death following PBI. However, mechanistic details about how alterations of Ca²⁺ levels contribute to SBI are not well characterized. In this paper, we first build a biophysical model for SBI related to calcium homeostasis (SBI-CH) to study the mechanistic details of PBI-induced disruption of CH, and how these disruptions affect the occurrence of SBI. Then, we construct a coupled SBI-CH model by formulating synaptic interactions to investigate how disruption of CH affects synaptic function and further promotes the propagation of SBI between neurons. Our model shows how the opening of voltage-gated calcium channels (VGCCs), decreasing of plasma membrane calcium pump (PMCA), and reversal of the Na⁺/Ca²⁺ exchanger (NCX) during and following PBI, could induce disruption of CH and further promote SBI. We also show that disruption of CH causes synaptic dysfunction, which further induces loss of excitatory-inhibitory balance in the system, and this might promote the propagation of SBI and cause neighboring tissue to be injured. Our findings offer a more comprehensive understanding of the complex interrelationship between CH and SBI.

NEW & NOTEWORTHY We build a mechanistic model SBI-CH for calcium homeostasis (CH) to study how alterations of Ca²⁺ levels following PBI affect the occurrence and propagation of SBI. Specifically, we investigate how the opening of VGCCs, decreasing of PMCA, and reversal of NCX disrupt CH, and further induce the occurrence of SBI. We also present a coupled SBI-CH model to show how disrupted CH causes synaptic dysfunction, and further promotes the propagation of SBI between neurons.

INTRODUCTION

Primary brain injury (PBI) is often defined as a sudden and profound injury to the brain. It happens and completes near the time of the inciting injury. But many people who are disabled and die as a result of PBI do not do so immediately. Secondary brain injury (SBI) refers to a new or worsening injury to the brain after an initial brain insult. SBI is a common and potentially preventable complication following many types of PBI [e.g., traumatic brain injury (TBI), stroke, subarachnoid hemorrhage (SAH), etc.] (1, 2). Mechanistically, a complex set of pathophysiologic alterations occur in SBI, including ischemia, cerebral hypoxia, excessive glutamate release, inflammation, and so on, in which the abnormal activities of various ions play key roles at the molecular level. A number of animal studies have indicated that abnormal potassium (K⁺) and sodium (Na⁺) ion-related activities contribute to exacerbating brain injury and SBI (2–5).

Besides K⁺ and Na⁺, alterations in Ca²⁺ signaling are also believed to play a role in SBI. It is reported that Ca²⁺ is one of the major culprits that contribute to the additional loss of neurons after PBI (6–8). However, the mechanistic details about how alterations of Ca²⁺ levels contribute to deteriorating PBI and further evolving into SBI are not well characterized. The purpose of this paper is to provide a neural computational model to study the mechanistic details of calcium homeostasis following PBI and how these alterations act in the occurrence and propagation of SBI from a microscopic modeling point of view.

The microscopic modeling approach is based on describing the detailed activities of a single neuron or a large number of interconnected neurons where each neuron is represented by an explicit neuron model at the cell level (9–11). It can be traced back to the 1950s, in which a single neuron model was proposed by Hodgkin and Huxley (12) to simulate nerve action potentials in the squid giant axon (say, HH model). Following Hodgkin and Huxley’s work, a large number of microscopic models have been developed and progressively used to simulate various electrophysiological activities and to shed light on the mechanisms underlying both neurophysiological and pathological phenomena (2).

In this paper, we first build a biophysical model for the SBI related to calcium homeostasis (SBI-CH) to better understand how alterations of calcium levels following PBI affect the occurrence of SBI. By incorporating both the dynamics of factors that increase calcium levels and decrease calcium levels, we extend the SBI model formalism proposed in our previous work (2), to model the role of calcium homeostasis in the evolution from PBI to SBI. The neurophysiological factors affecting Ca²⁺ levels we focus on in this work are included as voltage-gated calcium channels (VGCCs), plasma membrane calcium pump (PMCA), and Na⁺/Ca²⁺ exchanger (NCX), which are coupled to calcium ion dynamics, calcium reversal potential, tissue oxygen levels, and other ion dynamics. Figure 1 shows our conceptual model of SBI-CH. Then, we construct a coupled SBI-CH model incorporating connections between neurons, to investigate how disruption of CH affects synaptic function and further promotes the propagation of SBI. By modeling excitatory and inhibitory synaptic dynamics, two SBI-CH mechanistic models, with one for an excitatory neuron and another for an inhibitory neuron, are coupled through synaptic interactions.

Figure 1.

Conceptual model for how voltage-gated calcium channel (VGCC), plasma membrane calcium pump (PMCA), and Na⁺/Ca²⁺ exchanger (NCX) disrupt calcium homeostasis following primary brain injury (PBI) to induce secondary brain injury (SBI).

MATERIAL AND METHOD

In this section, we first summarize pathological alterations underlying calcium homeostasis (CH) related to brain injury. Then, two new biophysical models for SBI related to CH, a single SBI-CH model and a coupled SBI-CH model, are proposed by integrating the mathematical characterizations of CH into the existing SBI model.

Pathological Alterations of CH in Brain Injury

Prior researches indicate that major mechanisms for maintaining neuronal CH are through several types of voltage-gated Ca²⁺ channels (VGCCs), plasma membrane Ca²⁺-ATPase (PMCA), and Na⁺/Ca²⁺ exchanger (NCX) (7, 13, 14). Under normal conditions, all of these mechanisms operate together to keep the ratio of intracellular Ca²⁺ concentration and extracellular Ca²⁺ concentration at a relatively stable level (7). Increasing evidence indicates that disrupted CH related to pathological alteration of VGCCs, PMCA, and NCX often occurs following brain injury.

Destroyed CH related to VGCCs.

Prior studies indicate that [Ca²⁺]_i is elevated after PBI due to excessive activation or opening of VGCCs, which is believed to activate several degradative enzymes, setting in motion a cascade of events that eventually leads to secondary neuronal damage and death (13, 15). In an initial in vitro study, it has been shown that application of nimodipine (a typical L-type VGCC antagonist) immediately before a mechanical strain significantly reduced the average accumulation of [Ca²⁺]_i, and lowered the total number of neurons experiencing significant increases in [Ca²⁺]_i (16). In the work by Gurkoff et al. (17), both L-type and N-type VGCC antagonists successfully reduced calcium loads as well as neuronal and astrocytic cell death following mechanical injury. In vitro models of TBI, L- and N-type voltage-gated calcium channel blockers reduced [Ca²⁺]_i accumulation and glutamate release resulting in reduced cell death (15). In addition, it has been suggested that VGCC-blocking drugs could be neuroprotective in stroke (18). Also, T-type channels have been shown to be another class of VGCCs that may play a role in ischemic neuronal cell death (19).

Destroyed CH related to PMCA.

Several lines of evidence show that, as the major plasma membrane Ca²⁺ extruding system, the decreased or blockade of PMCA activity significantly contributes to cell injury and death with the accumulation of [Ca²⁺]_i. An experimental study in dogs with subarachnoid hemorrhage (SAH) suggested that the long-lasting and early occurrence of a decrease in PMCA expression induced a persistent disturbance of Ca²⁺ homeostasis. This indicates that damage to the plasma membrane in cerebral arterial smooth-muscle cells proceeds to myonecrosis after SAH (20). Schwab et al. (21) also suggested that disturbance of Ca²⁺ handling can be prevented by the expression of noncleavable PMCA, which markedly delays secondary cell necrosis. In addition, during glutamate-mediated neuronal death, PMCAs can be internalized, which contributes to Ca²⁺ deregulation and further supports a fundamental role for Ca²⁺ transporters in neuronal demise (6).

Destroyed CH related to NCX.

As a calcium extrusion system, the NCX exists in two modes: forward (Ca²⁺ exit) mode and reverse (Ca²⁺ entry) mode. A growing body of studies has shown that the reversal mode of NCX contributes to neuronal damage and a buildup of Ca²⁺ inside the neuron. Bano et al. (22) verified that proteolysis of NCX isoform 3 by calpains played a prominent role in excitotoxic Ca²⁺ elevation leading to neuronal demise. In the work by Ransom et al. (23), it has been reported that NCX blockers (e.g., bepridil, benzamil, and dichlorobenzamil) greatly protected the optic nerve from anoxic injury. Besides, Nicotera reported that loss of calcium extrusion mechanisms (e.g., NCX or PMCA) might cause necrosis in severe ischemia or apoptosis (24).

An Overview of SBI Model

In a previous work, Song et al. (2) proposed a biophysical model for the metabolic supply-demand mismatch hypothesis of SBI (SBI model). In that study, three mechanisms including decreased oxygen, increased extracellular potassium, and increased excitotoxicity are mathematically characterized and added to the classical Hodgkin–Huxley model.

The dynamic equations of the SBI model are formulated as follows:

$$\begin{array}{l}C\frac{\text{d}V}{\text{d}t}=I_{\text{ext}}-I_{\text{Na}}-I_{\text{K}}-I_{\text{Cl}}+I_{\text{exc}},\\ I_{\text{Na}}=G_{\text{Na}}{m}^{3}h(V-E_{\text{Na}})+G_{\text{NaL}}(V-E_{\text{Na}}),\\ I_{\text{K}}=G_{\text{K}}{n}^{4}h(V-E_{\text{K}})+G_{\text{K}}L(V-E_{\text{K}}),\\ I_{\text{Cl}}=G_{\text{ClL}}(V-E_{\text{Cl}}),\\ I_{\text{exc}}=g_{\text{excite}}(t)(E_{\text{excite}}-V),\\ \frac{\text{d}{g}_{\text{excite}}}{\text{d}t}=\frac{-(g_{\text{excite}}-g_{\text{mean}})}{{\text{τ}}_{\text{excite}}}+\sqrt{D_{\text{excite}}}{\text{ζ}}_{\text{excite}}.\end{array}$$ (1)

Here, the reversal potentials E_K, E_Na, E_Cl of K⁺, Na⁺, Cl⁻ are characterized by the ratio of extracellular and intracellular ion concentrations, that is,

$$\begin{array}{l}{E}_{\text{K}}=26.64\ln ({[{\text{K}}^{+}]}_{\text{o}})/{[{\text{K}}^{+}]}_{\text{i}}),\\ E_{\text{Na}}=26.64\ln ({[{\text{Na}}^{+}\text{]}}_{\text{o}})/{[{\text{Na}}^{+}\text{]}}_{\text{i}},\\ E\text{Cl}=26.64\ln ({[{\text{Cl}}^{-}\text{]}}_{\text{i}}/{[{\text{Cl}}^{-}\text{]}}_{\text{o}}),\end{array}$$

which obey the Nernst equation (12).

With the consideration of three mechanisms, the extracellular and intracellular ion concentrations are respectively expressed by

$$\begin{array}{l}\frac{\text{d}{[{\text{K}}^{+}\text{]}}_{\text{o}}}{\text{d}t}=\text{γβ}{I}_{\text{K}}-2\text{β}{I}_{\text{pump}}-I_{\text{diff}}-I_{\text{glia}}-2I_{\text{gliapump}},\\ {[{\text{K}}^{+}\text{]}}_{\text{i}}={[{\text{K}}^{+}\text{]}}_{\text{i,rest}}+({[{\text{Na}}^{+}\text{]}}_{\text{i,rest}}-{[{\text{Na}}^{+}\text{]}}_{\text{i}}),\\ \frac{\text{d}{[{\text{Na}}^{\text{+}}\text{]}}_{\text{i}}}{\text{d}t}=-\text{γβ}{I}_{\text{Na}}-3I_{\text{pump}},\\ {[{\text{Na}}^{+}]}_{\text{o}}={[{\text{Na}}^{+}\text{]}}_{\text{o,rest}}-\text{β}[{\text{Na}}^{+}{]}_{\text{i}}-[{\text{Na}}^{+}{\text{]}}_{\text{i,rest}}),\\ [{\text{Cl}}^{-}{]}_{\text{o}}\hspace{0.17em}\hspace{0.17em}\triangleq \hspace{0.17em}\hspace{0.17em}\text{constant},\\ {[{\text{Cl}}^{-}]}_{\text{i}}\hspace{0.17em}\hspace{0.17em}\triangleq \hspace{0.17em}\hspace{0.17em}\text{constant},\end{array}$$

where the formulation of currents I_pump, I_glia, I_gliapump, and I_diff are given in Eq. 2. A detailed explanation of the biological interpretation of all symbols in aforementioned formulas can be found in Ref. 2.

$$\begin{array}{l}{I}_{\text{pump}}=\frac{\text{ρ}}{(1+[\exp (25-{[{\text{Na}}^{\text{+}}\text{]}}_{\text{i}})/3]}\times \frac{1}{1+\exp (5.5-{[{\text{K}}^{+}]}_{\text{o}})},\\ I_{\text{glia}}=\frac{G_{\text{glia}}}{1+\exp ((18-{[{\text{K}}^{+}\text{]}}_{\text{o}})/2.5)},\\ I_{\text{gliapump}}=\frac{1}{3}\frac{\text{ρ}}{(1+[\exp (25-{[{\text{Na}}^{\text{+}}\text{]}}_{\text{i}})/3]}\times \frac{1}{1+\exp (5.5-{[{\text{K}}^{+}\text{]}}_{\text{o}})},\\ I_{\text{dif}f}={\mathrm{ϵ}}_{\mathrm{\text{k}}}({[{\text{K}}^{+}\text{]}}_{\text{o}}-{[{\text{K}}^{+}\text{]}}_{\text{bath}}),\\ \text{ρ}=\frac{{\text{ρ}}_{\text{max}}}{1+\exp [((20-{[{\text{O}}_2\text{]}}_{\text{o}}))/3]},\\ \frac{\text{d}{[{\text{O}}_2\text{]}}_{\text{o}}}{\text{d}t}=-\text{αλ}(I_{\text{pump}}+I_{\text{gliapump}})+{\mathrm{ϵ}}_0({[O_2]}_{\text{bath}}-{[{\text{O}}_2\text{]}}_{\text{o}}).\end{array}$$ (2)

SBI Model Related to Calcium Homeostasis

In this subsection, we first build a biophysical model of calcium homeostasis related to SBI (SBI-CH) for a single neuron, which focuses on explaining mechanistic details for the occurrence of SBI due to the disruption of CH. We then construct a coupled SBI-CH model aiming to investigate the propagation of SBI between neurons.

Single SBI-CH model.

Dynamics of VGCCs.

According to the discussion earlier, L-type, N-type, T-type VGCCs are believed to play prominent roles in regulating calcium homeostasis following brain injury. Therefore, three types of voltage-related currents I_Ca,L, I_Ca,N, I_Ca,T will be considered in this work. According to Ref. 25, we formulate them in an Ohm’s law-like fashion as follows,

$$\begin{array}{l}{I}_{\text{Ca,T}}=g_{\text{Ca,T}}{d}_{\text{T}}{f}_{\text{Ca,T}}(V-E_{\text{Ca}}),\\ I_{\text{Ca,N}}=g_{\text{Ca,N}}{d}_{\text{N}}{f}_{\text{Ca,N}}(V-E_{\text{Ca}}),\\ I_{\text{Ca,L}}=g_{\text{Ca,L}}{d}_{\text{L}}{f}_{\text{Ca,L}}(V-E_{\text{Ca}}),\end{array}$$ (3)

where g_Ca,T, g_Ca,N, g_Ca,L represent the corresponding maximal conductance. Similarly, the reversal potential E_Ca of Ca²⁺ is calculated by the Nernst equation as

$$E_{\text{Ca}}=13.32\ln \hspace{0.17em}({[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}/{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{o}}),$$

where [Ca²⁺]_i and [Ca²⁺]_o represent the intracellular and extracellular concentration of Ca²⁺.

Next, we explain the expressions of activation gating variables d_(·) and inactivation gating variables f_(Ca,·) for three types of calcium channels in Eq. 3. According to Landry and Canavier (25), the dynamics are described by

$$\begin{array}{l}\frac{\text{d}{d}_{\text{T}}}{\text{d}t}=\frac{\frac{1}{1+\exp (-(V+63.5)/1.5)}-d_{\text{T}}}{{\text{τ}}_{d_{\text{T}}}},\\ \frac{\text{d}{d}_{\text{N}}}{\text{d}t}=\frac{\frac{1}{1+\exp (-(V+45)/7)}-d_{\text{N}}}{{\text{τ}}_{d_{\text{N}}}},\\ \frac{\text{d}{d}_{\text{L}}}{\text{d}t}=\frac{\frac{1}{1+\exp (-(V+50)/20)}-{\text{d}}_L}{{\text{τ}}_{d_{\text{L}}}},\\ \frac{\text{d}{f}_{\text{Ca,T}}}{\text{d}t}=\frac{\frac{1}{1+\exp ((V+76.2)/3)}-f_{\text{T}}}{{\text{τ}}_{f_{\text{T}}}},\\ f_{\text{Ca,N}}=\frac{K_{\text{M,fCaN}}}{K_{\text{M,fCaN}}+{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}},\\ f_{\text{Ca,L}}=\frac{K_{\text{M},\text{fCaL}}}{K_{\text{M},\text{fCaL}}+{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}},\end{array}$$

where K_M,fCaN, K_M,fCaL are constant parameters, and τ_(·) denotes time constants satisfying

$$\begin{array}{l}{\text{τ}}_{d_{\text{T}}}=65\exp (-\frac{{(V+66)}^2}{40})+3.5,\\ {\text{τ}}_{f_{\text{T}}}=50\exp (-\frac{{(V+72)}^2}{100})+10,\\ {\text{τ}}_{d_{\text{N}}}=18\exp (-\frac{{(V+70)}^2}{5})+0.3,\\ {\tau }_{d_L}=18\exp (-\frac{{(V+45)}^2}{400})+1.5.\end{array}$$

Dynamics of PMCA.

This part focuses on characterizing the pump-related current I_Ca,P of PMCA. It is well known that neuronal pump activity is closely related to ATP levels, which in turn depends on oxygen levels (2). Different from our previous work in which consumption of [O₂]_o was only correlated with I_pump and I_gliapump, here, the ATP-related pump current I_Ca,P is supplemented by a term accounting for Ca²⁺ when modeling the dynamics of [O₂]_o, that is,

$$\begin{array}{l}\frac{\text{d}{[{\text{O}}_2\text{]}}_{\text{o}}}{\text{d}t}=-\text{α}(I_{\text{pump}}+I_{\text{gliapump}}+I_{\text{Ca,P}})\\ +{\mathrm{ϵ}}_0({[{\text{O}}_2\text{]}}_{\text{bath}}-{[{\text{O}}_2\text{]}}_{\text{o}}).\end{array}$$ (4)

In Eq. 4, the first term represents oxygen consumption caused by the Na⁺-K⁺ pump, glia pump, and Ca²⁺ pump, and the second term represents the oxygen supply caused by the microvasculature in vivo or bath solution in vitro ([O₂]_bath). Parameter α is a conversion factor converting pump current to oxygen concentration change, and ϵ₀ is the oxygen diffusion rate.

Next, we characterize the strength of the calcium pump ρ_Ca,P by considering the influence of oxygen. Different from the work in Ref. 25 where ρ_Cap is constant, here, we model it to be a sigmoid function of extracellular oxygen levels [O₂]_o, that is,

$${\text{ρ}}_{\text{Cap}}=\frac{{\text{ρ}}_{\text{Cap,max}}}{1+\exp ((20-{[{\text{O}}_2\text{]}}_{\text{o}})/3)},$$ (5)

where ρ_Cap,max is a constant denoting the maximum value of ρ_Ca,P.

With the above considerations, the current I_Ca,P of PMCA can be formulated as in Ref. 25 as follows,

$$I_{\text{Ca,P}}={\rho }_{\text{Cap}}\frac{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{({[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}+K_{\text{m,Cap}})},$$ (6)

where K_m,Cap is a constant denoting the value for the half-activation calcium concentration for I_Ca,P.

Dynamics of NCX.

This part considers the exchanger-related current I_NaCa of NCX based on Ref. 26, which is formulated by

$$\begin{array}{l}{I}_{\text{NaCa}}=I_{\text{NaCa,max}}*\frac{{[{\text{Na}}^{\text{+}}\text{]}}_{\text{o}}^3{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}{f}_{\text{1,NaCa}}}{(1+k_{\mathrm{sat}}{f}_{\text{1,NaCa}})F_{\text{NaCa}}}\\ -\frac{{{\text{[Na}}^{\text{+}}\text{]}}_{\text{i}}^3{{\text{[Ca}}^{\text{2+}}\text{]}}_o{f}_{2,\text{NaCa}}}{(1+k_{\mathrm{sat}}{f}_{1,\text{NaCa}})F_{\text{NaCa}}},\\ F_{\text{NaCa}}=(K_{\text{m,Nai}}^{\text{3}}+{[{\text{Na}}^{+}\text{]}}_{\text{o}}^3)*(K_{\text{m,Cai}}+{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{o}}),\\ f_{\text{1,NaCa}}=\exp (({\text{γ}}_0-1).*V*F./(R*T)),\\ f_{2,\text{NaCa}}=\exp ({\text{γ}}_0.*V*F./(R*T)).\end{array}$$ (7)

Here, I_NaCa,max is a scaling factor of I_NaCa; k_sat is the saturation factor of I_NaCa at very negative potentials; K_m,Nai and K_m,Cai are half-saturation concentration of the sodium channel and calcium channel, respectively; γ₀ is the position of the energy barrier controlling voltage dependence of I_NaCa; R is gas constant; T is the absolute temperature.

It should be emphasized that NCX has a direct bearing on the intracellular concentration of Na⁺. Accordingly, we remodel the dynamics of [Na⁺]_i by adding I_NaCa in Eq. 2, that is,

$$\frac{{\text{d[Na}}^{\text{+}}{]}_{\text{i}}}{\text{d}t}=-\text{γβ}{I}_{\text{Na}}-3I_{\text{pump}}+2I_{\text{NaCa}}+I_{\text{diff,pip}}.$$ (8)

Note that an additional current I_diff,pip = ζ([Na⁺]_pip−[Na⁺]_i) is also joined in characterizing the dynamics of [Na⁺]_i in Eq. 8, which represents the diffusion of Na⁺ from the perfusion pipette sodium concentration ([Na⁺]_pip) to the intracellular space in a slice preparation, and ζ is diffusion rate.

Mathematical formulation of SBI-CH model.

In this part, we first clarify the formulation of [Ca²⁺]_o and [Ca²⁺]_i, which are critical in the mathematical modeling of SBI-CH model.

On one hand, following methods for modeling calcium dynamics underlying burst firing oscillations in Ref. 27, we suppose [Ca²⁺]_o to be constant in our work. On the other hand, [Ca²⁺]_i is mainly affected by VGCCs for rising [Ca²⁺]_i level and PMCA for lowering [Ca²⁺]_i level (7). In addition, NCX also affects the change of Ca²⁺, which extrudes Ca²⁺ out of the cell under normal conditions, and reverses its operation to transport Ca²⁺ into the cell in certain pathological conditions (14). Therefore, in our single compartment model, we change the description of [Ca²⁺]_i dynamics in Ref. 28 where [Ca²⁺]_i is only related to VGCCs and PMCA, and remodel it as

$$\begin{array}{l}\frac{\text{d}{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{\text{d}t}=\frac{2f_{\text{Ca}}}{d_{\text{s}}F}[(I_{\text{Ca,T}}+I_{\text{Ca,N}}+I_{\text{Ca,L}})\\ -(I_{\text{Cap}}+I_{\text{NaCa}})].\end{array}$$ (9)

Here, f_Ca denotes the fraction of free calcium being a constant and F is Faraday’s constant.

The mathematical equations of our SBI-CH model are obtained by combining the SBI model with the formulation of mechanisms related to CH shown in Eqs. 3–9. The final model output is the membrane potential V, whose dynamic equation is

$$\begin{array}{l}C\frac{\text{d}V}{\text{d}t}=I_{\text{ext}}-I_{\text{Na}}-I_{\text{K}}-I_{\text{Cl}}-I_{\text{Ca,T}}\\ -I_{\text{Ca,N}}-I_{\text{Ca,L}}+I_{\text{exc}}.\end{array}$$ (10)

Coupled SBI-CH model.

Substantial evidence shows that brain injury often disrupts synaptic function. For example, Witkowski and coworkers (29) found that moderate effects on excitatory or inhibitory synaptic function at 48 h after TBI and a robust increase in excitatory inputs in slices prepared 1 h after injury. Hofmeijer and van Putten (30) showed that mild or moderate cerebral ischemia resulted in isolated, but persistent, synaptic failure. Moreover, synaptic damage and an effect of astrocytes associated with the protection of the brain from synaptic damage in ischemic stroke were depicted in Ref. 31.

To account for how brain injury impacts synaptic interactions and further promotes the propagation of SBI, we construct a coupled SBI-CH model. The coupled model consists of two neurons (one excitatory and one inhibitory), coupled through synaptic interactions (shown in Fig. 2). Both cells are modeled via the single SBI-CH model with the same parameter setting approaches.

Figure 2.

The structure of coupled secondary brain injury related to calcium homeostasis (SBI-CH) model. NCX, Na⁺/Ca²⁺ exchanger; PBI, primary brain injury; PMCA, plasma membrane calcium pump; VGCC, voltage-gated calcium channel

The membrane potentials of the excitatory neuron (denoted by V_e) and inhibitory neuron (denoted by V_i) are modeled respectively as

$$\begin{array}{l}C\frac{\text{d}{V}_{\text{e}}}{\text{d}t}=I_{\text{ext,e}}-I_{\text{Na,e}}-I_{\text{K,e}}-I_{\text{Cl,e}}-I_{\text{CaT,e}}-I_{\text{CaN,e}}\\ -I_{\text{CaL,e}}-I_{\text{syn,ie}}+I_{\text{exc,e}},\end{array}$$ (11)

$$\begin{array}{l}C\frac{\text{d}{V}_i}{\text{d}t}=I_{\text{ext,i}}-I_{\text{Na,i}}-I_{\text{K,i}}-I_{\text{Cl,i}}-I_{\text{CaT,i}}-I_{\text{CaN,i}}\\ -I_{\text{CaL,i}}-I_{\text{syn,ei}}+I_{\text{exc,i}},\end{array}$$ (12)

where I_syn,ei is the synaptic current from the excitatory neuron to the inhibitory neuron, and I_syn,ie is the synaptic current from the inhibitory neuron to the excitatory neuron. Both are modeled according to the work of Schiff and Ullah (32) as follows:

$$\begin{array}{l}{I}_{\text{syn,ie}}={\text{ω}}_{\text{ie}}{G}_{\text{ie}}{S}_{\text{ie}}\exp (-{\text{χ}}_{\text{ie}}/5))(V_{\text{e}}-E_{\text{in}}),\\ {\text{τ}}_{\text{ie}}\frac{\text{d}{S}_{\text{ie}}}{\text{d}t}=\frac{20}{1+\exp (-(V_{\text{i}}+20)/3)}*(1-S_{\text{ie}})-S_{\text{ie}},\\ {\text{τ}}_{\text{ei}}\frac{\text{d}{S}_{\text{ei}}}{\text{d}t}=\frac{20}{1+\exp (-(V_{\text{e}}+20)/3)}*(1-S_{\text{ei}})-S_{\text{ei}},\\ \frac{{\text{dχ}}_{\text{ie}}}{\text{d}t}={\text{η}}_{\text{ie}}(V_{\text{i}}+50)-0.4{\text{χ}}_{\text{ie}},\\ \frac{{\text{dχ}}_{\text{ei}}}{\text{d}t}={\text{η}}_{\text{ei}}(V_{\text{e}}+50)-0.4{\text{χ}}_{\text{ei}}.\end{array}$$

$$I_{\text{syn,ei}}={\omega }_{\text{ei}}{G}_{\text{ei}}{S}_{\text{ei}}\exp (-{\text{χ}}_{\text{ei}}/5))(V_{\text{i}}-E_{\text{ex}}),$$

The variables S_ei and S_ie give the temporal evolution of the synaptic input from excitatory neuron to inhibitory neuron, and from inhibitory neuron to excitatory neuron respectively; χ_ie and χ_ei represent the firing interplay between excitatory and inhibitory neurons; G_ei and G_ie are excitatory and inhibitory synaptic conductances; E_in and E_ex are reversal potentials of inhibitory and excitatory synapses; τ_ei and τ_ie are time constants for the excitatory and inhibitory synapses.

RESULTS

Experimental Approach

To investigate how alterations of calcium concentrations following PBI affect the occurrence and propagation of SBI in simulations, we here define two indexes, the SBI-severity index and the CH-disruption index, which are applied to measure the severity degree of SBI and the disruption degree of CH, respectively.

As stated in work of Song et al. (2), the cumulative burden of hypoxia is taken as a measure of severity and extent of SBI, so that the SBI-severity index (denoted SBII) is defined as

$$\text{SBII}={\int }_0^T\Theta ({\text{Thr}}_1-{[\text{O2}]}_{\text{o}})({\text{Thr}}_1-{[\text{O2}]}_{\text{o}})\text{d}t.$$ (13)

Here, Θ(·) represents the Heaviside function and Thr₁ is the SBI threshold whose value keeps a relatively low oxygen level. The higher the value of “SBII,” the more the cumulative burden of hypoxia will be, which implies a more severe SBI.

We then define the CH-disruption index (denoted CHI) as

$$\text{CHI}={\int }_0^T(\frac{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{o}}}(t)-{\text{Thr}}_2)\text{d}t,$$ (14)

where Thr₂ is the CH-threshold denoting the ratio of resting values between [Ca²⁺]_i and [Ca²⁺]_o, and T is the duration of the experiment. The higher the value of “CHI,” the more the deviation $\frac{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{o}}}$ from its resting value will be, which indicates more CH disruption occurring.

Moreover, it should be noted that seizures or seizure-like activity are common in these simulations and mirror seizures generated following acute brain injury (33, 34). Similar to the simulations in Ref. 2, we also define the model output as showing a “seizure” where there is a cluster of discharges.

The nominal values of all parameters in our proposed model are listed in Table 1 (see Refs. 25, 28, 35, and 36). In particular, the CH-threshold Thr₂ is set to be the ratio of resting [Ca²⁺]_i level to resting [Ca²⁺]_o level, whose values are ∼2.4 mM and 1.8 × 10^–4 mM (36). Therefore, ${\text{Thr}}_{\text{2}}=\frac{1.8}{2.4}\times {10}^{-4}\text{ mM}=0.75\times {10}^{-4}\text{ mM}$. The SBI-threshold Thr₁ is set to be 28 mg/L as in our previous work (2).

Table 1.

Nominal values of all parameters in SBI-CH model

Parameter	Values (Units)	Parameter	Values (Units)	Parameter	Values (Units)
G Na	30 mS/cm2	ρmax	1.25 mM/s	[O2]bath,normal	32 mg/L
G NaL	0.01750 mS/cm2	C	1 μF/cm2	IINaCa	1,500 μA/μF
G K	25 mS/cm2	α	5.3 g/mol	γ0	0.35
G KL	0.05 mS/cm2	λ	1	F	96,520
G ClL	0.05 mS/cm2	ϵ0	0.17 s−1	R	1.987
E Cl	−81.9386 mV	G glia	8 mM/s	T	308.15
β	7	ϵk	0.33 s−1	k sat	0.27
γ	0.0445 mM/s/μA/cm2	E excite	0 mV	K m,Nai	0.3 mM/L
[K+]bath	4 mM	I e	0.5	K m,Cai	7 mM/L
ωei	0.05	[Na+]i,rest	18 mM	f Ca	5
[K+]i,rest	140 mM	[Na+]i,rest	144 mM	d s	15
Thr	28 mg/L	Thr1	10–4 mM	ρCap,max	31.2
E ex	0 mV	E in	–80 mV	D excite	1
τexcite	3.3 ms	K M ,f CaN	0.0001	K M ,f CaL	0.00045
K m,Cap	0.0005	ρCap,max	31.2	ζ	1
[Na+]pip	25 mM	[K+]bath	4.5 mM	G ei	0.22
G ie	0.12	τei	20	τie	8
ηie	0.4	ηei	0.4	ωie	0.08

SBI-CH, secondary brain injury related to calcium homeostasis.

In the following, we apply the proposed biophysical models and defined two indices to investigate the effects of three factors (VGCCs, PMCA, and NCX) on CH disruption and how these alterations promote the occurrence and propagation of SBI. To this end, two key model parameters: [Na⁺]_pip and [O₂]_bath are varied in our experiments, which are closely related to the activities of VGCCs, PMCA, and NCX. Meanwhile, the explanations for several important observations obtained in the experiments will be also clarified by the proposed models. The simulations and programs for analysis were coded in the MATLAB programming language, and the fourth-order Runge–Kutta method was applied for integrating all differential equations. For Eq. 10, a noise realization was added at each step.

Experimental Results

SBI induced by opening of VGCCs.

From the foregoing discussion, it is known that the VGCCs play a major route leading to the accumulation of [Ca²⁺]_i after brain injury. Here, we investigate how disrupted CH, caused by the excessive opening of VGCCs, may promote the occurrence of SBI.

Multiple studies demonstrate that brain injury facilitates abnormal Na⁺ influx through mechanically sensitive Na⁺ channels, which triggers membrane depolarization and subsequently induces the opening of various VGCCs (37). To mimic the effects of increasing [Na⁺]_i on VGCCs, we increase the value of parameter [Na⁺]_pip in its physiological range (38). Here, we set [Na⁺]_pip to be 25 mM, 28 mM, and 33 mM, respectively, and other parameters are set to be their nominal values. The corresponding results are shown in Fig. 3.

Figure 3.

The disrupted calcium homeostasis, caused by opening of voltage-gated calcium channels (VGCCs), promotes the epileptiform discharges and secondary brain injury (SBI). In each subfigure, plots show intracellular Na⁺ ([Na⁺]_i), calcium currents corresponding to N-, L-, T-type channels (I_CaN, I_CaL, I_CaT), current of NCX (I_NaCa), the ratio of intracellular and extracellular calcium concentration $\left(\frac{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{o}}}\right)$, intracellular and extracellular potassium concentration ([K⁺]_i and [K⁺]_o), membrane potential (V), and extracellular oxygen concentration ([O₂]_o). These are shown for three levels of pipette sodium: [Na⁺]_pip = 25 mM (I), [Na⁺]_pip = 28 mM (II), and [Na⁺]_pip = 33 mM (III). The area of the gray regions represent SBI-severity index (SBII) and calcium homeostasis-disruption index (CHI) defined in Eqs. 13 and 14, respectively.

From Fig. 3, I and II, we can see that as [Na⁺]_pip increases, [Na⁺]_i increases (shown in Fig. 3IIa). And when large amounts of Na⁺ fluxes into the cell, which activates the opening of calcium channels, the calcium activities corresponding to T-type, N-type, and L-type channels are increased (i.e., I_CaT, I_CaN and I_CaL, shown in Fig. 3II, b–d). The opening of VGCCs promotes the accumulation of [Ca²⁺]_i. Then we plot the curve of $\frac{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{o}}}$, as shown in Fig. 3IIf. We also observe that it significantly deviates from its resting values (see the blue dotted line in the third row), where the gray area represents the CH index with the value of 0.0058 (i.e., CHI = 0.0058). Furthermore, the damaged CH promotes epileptiform discharges (as shown in Fig. 3IIg), and causes the occurrence of SBI (shown in Fig. 3IIh). The blue dotted line represents the SBI threshold and the gray area corresponds to the SBI index with the value of 5.6856 (i.e., SBII = 5.6856).

From Fig. 3, II and III, we see that, as [Na⁺]_pip increases more, the activities corresponding to T-, N-, L-type calcium channels are further intensified, which is followed by the largely disrupted CH (i.e., CHI = 0.0071). Then it promotes increased seizures and induces greater SBI (i.e., SBII = 25.2668). We also found that the increase of [Na⁺]_pip not only increases the number of seizures but also prolongs the duration of seizures.

Figure 4A shows the bifurcation plot of [Ca²⁺]_i with the parameter [Na⁺]_pip, where the maximum and minimum of local [Ca²⁺]_i are presented by the blue and red curves. Membrane potential traces for three values of [Na⁺]_pip = 25, 30, and 38 mM are shown in Fig. 4, B–D corresponding to three vertical lines in Fig. 4A. The neuron is in the inactivated state for low values of [Na⁺]_pip (see Fig. 4B). As [Na⁺]_pip increases, the membrane is depolarized and VGCCs is opened, resulting in a significant accumulation of [Ca²⁺]_i. A region of large slow [Ca²⁺]_i oscillations (see Fig. 4C, inset) appears forming a foundation for spontaneous seizures and SBI. When [Na⁺]_pip increases further, the neuron goes to an overactivated state and fires tonically producing large fast local [Ca²⁺]_i oscillations (Fig. 4D, inset), but not seizures.

Figure 4.

A: the maximum and minimum of [Ca²⁺]_i as a function of [Na⁺]_pip. B–D: membrane potential traces for the three regions of bifurcation diagram when [Na⁺]_pip equals 25 mM, 30 mM, and 38 mM, respectively, corresponding to three vertical lines in A.

SBI induced by decreasing of PMCA.

Here, we verify that the damaged CH, caused by decreasing of PMCA, may promote the occurrence of SBI.

As the name suggests, this pump requires energy, in the form of one ATP molecule for every Ca²⁺ extruded (39). When the supply of ATP to PMCA is reduced, the ability of cells that pump Ca²⁺ into the extracellular space is weakened, resulting in CH disruption. The decreased supply of ATP after initial brain injury is mainly due to damaged mitochondria or insufficient oxygen resulting from reduced cerebral blood flow. In this study, we consider the function of O₂ levels in producing ATP with the hypothesis that mitochondria are working normally. To model the reduced supply of ATP on PMCA, we decrease [O₂]_bath in Eq. 4 from its normal value of 32 32 mg/L to lower values. In addition, a small amount of external positive current (I_e = 0.5 µA/cm²) is injected into the neuron to mimic a slightly cellular excitable state with a normal [Na⁺]_pip. Here, we set [O₂]_bath to be 32 mg/L, 28 mg/L, and 26 mg/L, respectively, and other parameters are set to be their nominal values. The corresponding results are shown in Fig. 5.

Figure 5.

The disrupted calcium homeostasis, caused by decreasing of plasma membrane calcium pump (PMCA), promotes the secondary brain injury (SBI). Plots show current of PMCA (I_Ca,P), current of Na⁺/Ca²⁺ exchanger (NCX) (I_NaCa), the ratio of intracellular and extracellular calcium concentration $\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{{[{\text{Ca}}^{2+}\text{]}}_{\text{o}}}\right)$, membrane potential (V), extracellular oxygen concentration ([O₂]_o). These are shown for three levels of bath oxygen: [O₂]_bath = 32 mg/L (A), [O₂]_bath = 28 mg/L (B), and [O₂]_bath = 26 mg/L (C). The area of the gray regions represents SBI-severity index (SBII) and calcium homeostasis-disruption index (CHI).

We can see from Fig. 5A, with normal [O₂]_bath (i.e., 32 32 mg/L), the neuron exhibits tonic firing but not seizures. As [O₂]_bath decreases to 28 32 mg/L, we observe that the I_Ca,P decreases (shown in the first row of Fig. 5B). In this case, Ca²⁺ cannot be pumped out of the cell promptly so that the concentration [Ca²⁺]_i steadily accumulates. As a result, $\frac{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{o}}}$ significantly deviates from its resting value (shown in the third row of Fig. 5B) where CHI = 0.0587. In particular, it can be seen from the red dashed box in Fig. 5B that I_Ca,P has a descending trend, while $\frac{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{[{\text{Ca}}^{\text{2+}}\text{]}}_{\text{o}}}$ has an ascending trend. Simultaneously, the damaged CH promotes epileptiform discharges (shown in the fourth row of Fig. 5B), and further aggravates SBI (shown in the last row of Fig. 5B) where SBII = 420.0185.

When [O₂]_bath decreases more, the current I_Ca,P drops further, which induces intensifying damage of CH (i.e., CHI = 0.0759) and worsening severity of SBI (i.e., SBII = 614.7345) as shown in Fig. 5C. However, in this case, the seizure frequency decreases as SBI progresses (comparing with Fig. 5, B and C). This is because Na⁺-K⁺-pump becomes weaker with the decreasing oxygen level so that its ability to restore the sodium gradient declines, hence, the neuron will gradually move into an inactivated state.

SBI induced by reversal of NCX.

Here, we verify that disrupted CH, caused by the reversed mode of NCX, may promote the occurrence of SBI.

There are two ways to push NCX into operating in its reverse mode following the initial injury. The first way directly relates to [Na⁺]_i, where increasing [Na⁺]_i activates the reverse mode of NCX to uptake Ca²⁺ into the cell. As can be observed from Fig. 3, I–IIIe, as [Na⁺]_pip increases, the NCX transfers to its reverse mode, corresponding to a gradually significant negative current I_NaCa. In such a case, overall, the opening of VGCCs and the reversal of NCX jointly push the accumulation of [Ca²⁺]_i and damage CH together. The second way correlates with the [O₂]_o. As [O₂]_o decreases, the ability of Na⁺-K⁺-pump is limited, leading to higher [Na⁺]_i and [K⁺]_o, which depolarize the neuron and push the NCX switching from the forward mode to reverse mode. It can be observed from the second row of Fig. 5B that the reversing activities of NCX become stronger with the decreasing [O₂]_bath. Overall, the decreasing PMCA and reversal of NCX lead to the accumulation of [Ca²⁺]_i, working together to disrupt CH. When [O₂]_bath is too low to maintain the Na⁺-K⁺-pump for restoring the Na⁺ gradient, the neuron will gradually get into the inactivated state, and then, the activities of NCX become fine away.

The effects of both [Na⁺]_pip and [O₂]_o on CH and SBI are shown in Fig. 6. Membrane potentials simulated for 140-s duration were generated, and those simulated for the first 10 s were ignored to exclude transient effects. We see that the smaller the values of [O₂]_bath and the larger the values of [Na⁺]_pip are, the more serious CH is disrupted (i.e., larger the value of CHI), leading to the severer SBI (i.e., larger the value of SBII). The relationship between CHI and SBII is also illustrated in Fig. 7, where five lines corresponding to [Na⁺]_pip = 25,27,29,31,33 are depicted, respectively. We can see that there is a generally positive correlation between CH disruption and SBI. That is, SBI will occur as long as the value of CHI is large enough. Meanwhile, the larger the value of CHI, the more severe SBI will be.

Figure 6.

The effect of intracellular Na⁺ and oxygen levels on calcium homeostasis and brain damage. CHI, calcium homeostasis-disruption index; SBII, secondary brain injury-severity index.

Figure 7.

The relationship between disrupted calcium hemostasis and secondary brain injury. In each line, the [O₂]_bath is decreased from its normal value 32 32 mg/L to 23 32 mg/L. CHI, calcium homeostasis-disruption index; SBII, secondary brain injury-severity index.

Synaptic dysfunction and secondary brain injury.

In this subsection, to investigate how disruption of CH in one neuron affects synaptic function and further promotes the propagation of SBI between neurons, we use the same model for two cells (one excitatory cell and one inhibitory cell) and couple them through synaptic interactions. The results are shown in Fig. 8, where the left column corresponds to activities of the excitatory neuron (denoted $\mathbb{\text{E}}$), and the right column corresponds to the inhibitory neuron (denoted $\mathbb{I}$) under different ways of connecting the neurons (i.e., no connection, one-way connection from $\mathbb{E}$ to $\mathbb{I}$), bidirectional-connection). In each subfigure, the $\frac{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{o}}}$, membrane potential V, oxygen level [O₂]_o, and synaptic current I_syn are illustrated, respectively.

Figure 8.

The disruption of calcium homeostasis caused by brain injury affects synaptic function and further promotes secondary brain injury (SBI). In each subfigure, plots show the $\frac{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{i}}}{{{\text{[Ca}}^{\text{2+}}\text{]}}_{\text{o}}}$, membrane potential V, oxygen level [O₂]_o, and synaptic current I_syn. These are shown for three connection patterns between one excitatory neuron ($\mathbb{E}$) and one inhibitory neuron ($\mathbb{I}$): no connection between two neurons (A and D), one-way connection from $\mathbb{E}$ to $\mathbb{I}$ (B and E), and bidirectional-connection between two neurons (C and F).

We assume that SBI begins in the excitatory neuron. In what follows, we observe how SBI propagates from $\mathbb{E}$ to $\mathbb{I}$. Here, we set the [Na⁺]_pip in the excitatory neuron to be 27 mM to guarantee the occurrence of SBI in $\mathbb{E}$.

In the first case where no connection exists between two neurons (i.e., ω_ei = ω_ie = 0), there is no synaptic inputs for each neuron (i.e., I_syn,ei = I_syn,_ie = 0). It can be seen from Fig. 8, A and D that injury only occurs in the excitatory neuron (SBII_e = 2.7491), while nothing happens in the inhibitory neuron (SBII_i = 0).

In the second case where a one-way connection from $\mathbb{E}$ to $\mathbb{I}$ exists (i.e., ω_ei = 0.05, ω_ie = 0), there is a negative synaptic input for inhibitory neuron $\mathbb{I}$ (see last row of Fig. 8B). Hence, the “–I_syn,ei” will provide an excitatory stimulus to $\mathbb{I}$, which may promote seizures and SBI in $\mathbb{I}$ (see second and third rows of Fig. 8E). We have also verified that a similar phenomenon can be obtained if some disturbance ϵ is added to ω_ei, where $\mathrm{ϵ}\sim N({\text{μ}}_0,{\text{σ}}_0^2)$. We conducted a number of trials by randomly selecting ω_ei based on $N(0.05+{\text{μ}}_0,{\text{σ}}_0^2)$ (${\text{μ}}_0\in (0,0.2),{\text{σ}}_0^2\in (0,0.4)$), as well as ω_ei > 0. We found that excitatory stimulus from $\mathbb{E}$ can always promote seizures and SBI in $\mathbb{I}$.

In the third case where a bidirectional-connection between $\mathbb{E}$ and $\mathbb{I}$ exists (i.e., ω_ei = 0.05 ω_ie = 0.08), besides the I_syn,ei, there is an additional positive synaptic input I_syn,ie for excitatory neuron $\mathbb{E}$ (the last row of Fig. 8F). Hence, –I_syn,ie provides a decreased inhibitory stimulus to $\mathbb{E}$, which facilitates the excitability of $\mathbb{E}$ and may further exacerbate the seizures and SBI occurring in $\mathbb{E}$ (see second and third rows of Fig. 8C). In turn, the increased excitability in $\mathbb{E}$ also causes more seizures and SBI to occur in $\mathbb{I}$ through increased excitatory synaptic current (see second and third rows of Fig. 8F).

We also have conducted another experiment in which we assume that SBI begins in the inhibitory neuron; similar experimental results are obtained. All of these results indicate that the disrupted CH will cause synaptic dysfunction and further induce loss of excitatory-inhibitory balance in a system, which may further promote SBI and cause nearby tissue to be injured

SUMMARY AND DISCUSSION

In this paper, we have built a biophysical modeling simulation of brain injury to study mechanistic details about PBI-induced disruption of CH, and how this disruption affects the occurrence and propagation of SBI. By mathematically characterizing the dynamics of VGCCs, PMCA, and NCX, we first constructed the model SBI-CH by modeling the role of CH in the evolution from PBI to SBI. Then, we proposed a coupled SBI-CH model by modeling the synaptic interaction between excitatory and inhibitory neurons, to investigate how disruption of CH affects synaptic function and further promotes the propagation of SBI between neurons.

It is worth mentioning that a biophysical model of SBI has been proposed in Ref. 2. However, the modeling mechanisms in these two works are different. The model in Ref. 2 focuses on metabolic supply-demand mismatch related to SBI, where the dynamics of [K⁺]_o, [O₂]_o and excitatory synaptic current are added to the basic HH model. By contrast, our work presently focuses on CH related to SBI, where additional dynamics of VGCCs, PMCA, and NCX related to Ca²⁺ dynamics are added to the model of Ref. 2.

On one hand, we have verified that the physical insults to the brain induce membrane depolarization, which results in the opening of VGCCs and activates the reverse mode of NCX. Both of these promote the accumulation of [Ca²⁺]_i, and then cause CH to deviate from its resting value. The disrupted CH further exacerbates the extent of the injury. Such computational results are summarized as a mechanistic pathway for how PBI evolves into SBI in Fig. 9A.

Figure 9.

The mechanistic pathways from primary brain injury (PBI) to secondary brain injury (SBI). A: pathway I: The increasing [Na⁺]_i (resulted from the increase of parameter [Na⁺]_pip) activates the opening of voltage-gated calcium channels (VGCCs) and the reverse mode of Na⁺/Ca²⁺ exchanger (NCX), which promote disruption of calcium homeostasis and go on to induce occurrence of secondary brain injury (SBI). B: pathway II: The decreasing [O₂]_o (resulted from the decrease of parameter [O₂]_bath) limits ability of plasma membrane calcium pump (PMCA) and promotes NCX to its reverse mode, which promote the disruption of calcium homeostasis and go on to induce occurrence of SBI.

On the other hand, we have verified that the reduced supply of oxygen after PBI results in the limited ability of PMCA to pump Ca²⁺ out of the cell, and pushes NCX to uptake Ca²⁺ into the cell. Both of them cause accumulation of [Ca²⁺]_i, which also disrupts CH and exacerbates the extent of the injury. The corresponding results are summarized as a mechanistic pathway shown in Fig. 9B.

Our model can also be applied to explain some other interesting phenomena. Baburamani et al. (40) found that individuals with high capillary density are less vulnerable to hypoxia/ischemia insults than those with capillary rarefaction. And it is known that high capillary density often corresponds to more supply of oxygen and ATP (41). Therefore, our findings also can be used to explain why certain comorbidities (e.g., diabetes, advanced age, nutrient deficiencies), in which capillary rarefaction plays a pivotal role, enhance vulnerability to hypoxic/ischemic injury.

Furthermore, we have verified that disrupted CH can cause synaptic dysfunction and further induce the loss of excitatory-inhibitory balance, and this might promote the occurrence of SBI and induce nearby tissue to be injured. This is consistent with the findings of in vivo studies (29).

Our model can also be applied to explain some observations about Ca²⁺ and excitotoxicity, which correlates with SBI in work (2). For example, Kiedrowski (42) suggested that the load of [Ca²⁺]_i generally increases following glutamate excitotoxicity in vivo. From the standpoint of modeling, we increase the value of g_mean in Eq. 1 to mimic increasing excitotoxicity. Simulation results are shown in Fig. 10, A–C where g_mean = 1, 1.5, 2, respectively. It can be observed that as excitotoxicity increases (from left to right), the activities of the reverse mode of NCX intensify, leading to large amounts of Ca²⁺ influx into the neuron. Consequently, disruption of CH occurs, resulting in SBI. This is consistent with the finding that NCX reversal can result in both enhanced Ca²⁺ influx and excitotoxic cell death (43).

Figure 10.

The disrupted calcium homeostasis, caused by increasing of excitotoxicity, promotes the secondary brain injury (SBI). Plots show current of Na⁺/Ca²⁺ exchanger (NCX) (I_NaCa), the ratio of intracellular and extracellular calcium concentration $\left(\frac{{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}}{{[{\text{Ca}}^{2+}\text{]}}_{\text{o}}}\right)$, membrane potential (V), and extracellular oxygen concentration ([O₂]_o). These are shown for three levels of excitotoxicity corresponding to g_mean = 1, 1.5, 2 (A–C, respectively). The area of the gray regions represent SBI-severity index (SBII) and calcium homeostasis-disruption index (CHI).

It is necessary to clarify that the coupled model in our work consists of two neurons. More realistic network-level models with many interconnected neurons may provide additional insight for understanding the mechanisms underlying the occurrence and propagation of SBI. Therefore, in future work, network-level neural computational modeling is a research frontier we plan to follow. In that future work, more complicated neuron coupling patterns will be explored, including synaptic connection and ion diffusion.

GRANTS

Dr. Jiang-Ling Song and Dr. Rui Zhang were supported by the National Natural Science Foundation of China under Grants 12071369 and 62006189, the Natural Science Foundation of Shaanxi Province under Grant 2021JQ-430, and the Innovative Talents Promotion Plan of Shaanxi Province under Grant 2018TD-016. Dr. Westover is supported by the Glenn Foundation for Medical Research and the American Federation for Aging Research through a Breakthroughs in Gerontology Grant; through the American Academy of Sleep Medicine through an AASM Foundation Strategic Research Award; and by National Institutes of Health Grants 1R01NS102190, 1R01NS102574, 1R01NS107291, and 1RF1AG064312.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

J.-L.S., M.B.W., and R.Z. conceived and designed research; J.-L.S. performed experiments; J.-L.S. and R.Z. interpreted results of experiments; J.-L.S. prepared figures; J.-L.S. drafted manuscript; J.-L.S., M.B.W., and R.Z. edited and revised manuscript; R.Z. approved final version of manuscript.