Dynamics and conditions for inhibitory synaptic current to induce bursting and spreading depolarization in pyramidal neurons

Coupling model of a pyramidal neuron and an interneuron

To focus on the inhibitory synaptic current, as shown in Fig. 10a, a pyramidal neuron receiving inhibitory synaptic current from an inhibitory interneuron for the coupling model, with the excitatory synapse from the pyramidal neuron to the interneuron ignored, since it can be speculated that the excitatory synapse can enhance activity of the interneuron. Then, the excitatory synapse may facilitate the uncommon MMB, which will be studied in future.

Sketch map of the coupling model and the network model. (a) An inhibitory synapse (blue arrow) from the interneuron (left) to the pyramidal neuron (right). The over-expressed sodium (Na⁺) channel is labeled, and J_i represent the depolarization current for interneuron. J_e represent the depolarization current and \({I}_{\text{app}}\) denotes the negative pulse stimulation to the pyramidal neuron, and the black two-way arrows indicate the movement of potassium ions. \({\left[{\text{K}}^{+}\right]}_{\text{o}}\) stands for the extracellular potassium ion concentration; (b) Network composed of \(N\times N\) pyramidal neurons with K⁺ ion diffusion in the extracellular space between two adjacent neurons (lateral diffusion, two-way arrows). The red one (r, u) is suppressed by an inhibitory synapse from an interneuron.

Pyramidal neuron model

A pyramidal neuron model used in Ref²⁴ is employed and described as follows:

where \({V}_{\text{e}}\) denotes the membrane voltage, n and h are the gating variables of K⁺ channel and Na⁺ channel, respectively. \({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\), \({\left[{\text{K}}^{+}\right]}_{\text{o}}\), \({\left[{\text{K}}^{+}\right]}_{\text{i}}\), \({\left[{\text{Na}}^{+}\right]}_{\text{i}}\), and \({\left[{\text{Cl}}^{-}\right]}_{\text{i}}\) denotes the concentrations of intracellular calcium, extracellular K⁺, intracellular K⁺, intracellular Na⁺, and intracellular chloride (Cl^–), respectively. \({\left[{\text{K}}^{+}\right]}_{\text{o}}\), \({\left[{\text{K}}^{+}\right]}_{\text{i}}\) , \({\left[{\text{Na}}^{+}\right]}_{\text{i}}\), and \({\left[{\text{Cl}}^{-}\right]}_{\text{i}}\) are four slow variables, since τ = 1000 is very large. Then, fast-slow analysis to the MMB is difficult since multiple slow variables.

In Eq. (1), \({C}_{\text{me}}\) is the membrane capacitance, and \({J}_{\text{e}}\) means the depolarization current to enhance the firing activity. \({I}_{\text{Na}}\) and I_K are the Na⁺ and K⁺ current, respectively. \({I}_{\text{L}}\) represents the leak current, which is composed of sodium leak (\({I}_{\text{NaL}}\)), potassium leak (\({I}_{\text{KL}}\)), and chlorine leak (\({I}_{\text{ClL}}\)) currents. \({I}_{\text{NaP}}\), \({I}_{\text{AHP}}\), and \({I}_{\text{pump}}\) refer to the persistent sodium, calcium-activated potassium, and sodium–potassium pump currents, respectively. The detailed expressions of these currents in Eq. (1) are described as follows: \({I}_{\text{L}}={I}_{\text{NaL}}+{I}_{\text{KL}}+{I}_{\text{ClL}},\) \({I}_{\text{NaL}}={g}_{\text{NaL}}\left({V}_{\text{e}}-{E}_{\text{Na}}\right),\) \({I}_{\text{KL}}={g}_{\text{KL}}\left({V}_{\text{e}}-{E}_{\text{K}}\right),\) \({I}_{\text{ClL}}={g}_{\text{ClL}}\left({V}_{\text{e}}-{E}_{\text{Cl}}\right),\) \({I}_{\text{K}}={g}_{\text{K}}{n}^{4}\left({V}_{\text{e}}-{E}_{\text{K}}\right),\) \(I_{\text{Na}} = g_{\text{Na}} m_{\infty \text{Na}}^{3} h\left( {V_{\text{e}} – E_{\text{Na}} } \right),\) \(I_{\text{NaP}} = g_{\text{P}} m_{\infty \text{Na}}^{3} \left( {V_{\text{e}} – E_{\text{Na}} } \right),\) \({I}_{\text{AHP}}={g}_{\text{AHP}}{\left[{\text{Ca}}^{2+}\right]}_{\text{i}}/\left({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}+1\right)\left({V}_{\text{e}}-{E}_{\text{K}}\right),\) \({I}_{\text{pump}}={\rho }_{\text{pump}}/\left[1+\text{exp}\left(3.5-{\left[{\text{K}}^{+}\right]}_{\text{o}}\right)\right]/\left[1+\text{exp}\left(\left(22-{\left[{\text{Na}}^{+}\right]}_{\text{i}}\right)/3\right)\right]/\gamma,\)\(m_{\infty \text{Na}} = a_{m} /\left( {a_{m} + b_{m} } \right)\), \({a}_{m}=0.32\left({V}_{\text{e}}+54\right)/\left[1-\text{exp}\left(-\left({V}_{\text{e}}+54\right)/4\right)\right]\), \({b}_{m}=0.28\left({V}_{\text{e}}+27\right)/\left[\text{exp}\left(\left({V}_{\text{e}}+27\right)/5\right)-1\right]\), where \({g}_{\text{NaL}}\), \({g}_{\text{KL}}\), and \({g}_{\text{ClL}}\) are the conductance. \({E}_{\text{Na}}\), \({E}_{\text{K}}\), and \({E}_{\text{Cl}}\) denote the reversal voltage. \({g}_{\text{K}}\) and \({g}_{\text{Na}}\) represent the maximum conductance of I_K and \({I}_{\text{Na}}\), respectively, \(m_{\infty \text{Na}}\) is steady state of activated variable. \({g}_{\text{P}}\), \({g}_{\text{AHP}}\), and \({\rho }_{\text{pump}}\) denote the maximum conductance. γ = S/(F·Vol), where S and Vol denotes the surface and volume of the pyramidal neuron, and F is the Faraday constant.

In Eqs. (2), (3), n represents the activated gating variable of K⁺ channel, and h denotes the inactivated gating variable of Na⁺ channel. The functions in Eqs. (2), (3) are described as follows: \({a}_{n}=0.032\left({V}_{\text{e}}+52\right)/\left[1-\text{exp}\left(-\left({V}_{\text{e}}+52\right)/5\right)\right]\), \({b}_{n}=0.5\text{exp}\left(-\left({V}_{\text{e}}+57\right)/40\right)\), \({a}_{h}=0.128\text{exp}\left(-\left({V}_{\text{e}}+50\right)/18\right)\), and \({b}_{h}=4/\left[1+\text{exp}\left(-\left({V}_{\text{e}}+27\right)/5\right)\right]\).

In Eq. (4), \({g}_{\text{Ca}}\) and \(m_{\infty \text{Ca}}\) denote the maximum conductance and the steady state of activated variable of the calcium current \(g_{\text{Na}} m_{\infty \text{Na}} \left( {V_{\text{e}} – E_{\text{Na}} } \right)\), respectively, and \(m_{\infty \text{Na}} = 1/\left[ {1 + \exp \left( { – \left( {V_{\text{e}} + 25} \right)/2.5} \right)} \right]\). \({\tau }_{\text{Ca}}\) is the time constant of \({\left[{\text{Ca}}^{2+}\right]}_{\text{i}}\).

In Eqs. (5)–(8), \({I}_{\text{KCC}}\) is K⁺-Cl^– cotransporters and \({I}_{\text{NKCC}}\) is Na⁺-K⁺-Cl^– cotransporters. \({I}_{\text{diffKo}}\) and \({I}_{\text{diffKi}}\) simulate the potassium diffusion in the extra- and intra-cellular space, respectively. The detailed expressions of the ion currents related to Eqs. (5)–(8) are described as follows: \({I}_{\text{KCC}}=0.3\text{ln}\left({\left[{\text{K}}^{+}\right]}_{\text{i}}{\left[{\text{Cl}}^{-}\right]}_{\text{i}}/\left({\left[{\text{K}}^{+}\right]}_{\text{o}}{\left[{\text{Cl}}^{-}\right]}_{\text{o}}\right)\right)\), \({I}_{\text{NKCC}}=0.1\left[\text{ln}\left({\left[{\text{K}}^{+}\right]}_{\text{i}}{\left[{\text{Cl}}^{-}\right]}_{\text{i}}/\left({\left[{\text{K}}^{+}\right]}_{\text{o}}{\left[{\text{Cl}}^{-}\right]}_{\text{o}}\right)\right)+\text{ln}\left({\left[{\text{Na}}^{+}\right]}_{\text{i}}{\left[{\text{Cl}}^{-}\right]}_{\text{i}}/\left({\left[{\text{Na}}^{+}\right]}_{\text{o}}{\left[{\text{Cl}}^{-}\right]}_{\text{o}}\right)\right)\right]/\left[1+\text{exp}\left(16-{\left[{\text{K}}^{+}\right]}_{\text{o}}\right)\right]\), \({I}_{\text{diffKo}}=\left({\left[{\text{K}}^{+}\right]}_{\text{o}}-{\text{K}}_{\text{o}0}\right)/{\tau }_{\text{Ko}}\), and \({I}_{\text{diffKi}}=\left({\left[{\text{K}}^{+}\right]}_{\text{i}}-{\text{K}}_{\text{i}0}\right)/{\tau }_{\text{Ki}}\), where \({\tau }_{\text{Ko}}\) and \({\tau }_{\text{Ki}}\) represent the time constant of extra- and intra-cellular K⁺ diffusion, respectively, \({\text{K}}_{\text{o}0}\) denotes the potassium concentration of the bathing solution. \({\text{K}}_{\text{i}0}\) denotes the equilibrium concentration of intracellular K⁺. \({\left[{\text{Na}}^{+}\right]}_{\text{o}}=144 \text{mM}-\beta \left({\left[{\text{Na}}^{+}\right]}_{\text{i}}-18 \text{mM}\right)\), \({\left[{\text{Cl}}^{-}\right]}_{\text{o}}=130 \text{mM}-\beta \left({\left[{\text{Cl}}^{-}\right]}_{\text{i}}-6 \text{mM}\right)\), where 144 mM and 18 mM is the sodium concentration outside and inside the neuron, respectively, and 130 mM and 6 mM correspond to the normal resting \({\left[{\text{Cl}}^{-}\right]}_{\text{o}}\) and \({\left[{\text{Cl}}^{-}\right]}_{\text{i}}\), respectively. The Nernst potentials of Na⁺, K⁺, and Cl^– are given as follows: \({E}_{\text{Na}}=26.64\text{ln}\left({\left[{\text{Na}}^{+}\right]}_{\text{o}}/{\left[{\text{Na}}^{+}\right]}_{\text{i}}\right)\), \({E}_{\text{K}}=26.64\text{ln}\left({\left[{\text{K}}^{+}\right]}_{\text{o}}/{\left[{\text{K}}^{+}\right]}_{\text{i}}\right)\), and \({E}_{\text{Cl}}=26.64\text{ln}\left({\left[{\text{Cl}}^{-}\right]}_{\text{i}}/{\left[{\text{Cl}}^{-}\right]}_{\text{o}}\right)\).

The parameter values of the pyramidal neuron are as follows: \({C}_{\text{me}}\) = 1 μF/cm², \({g}_{\text{NaL}}\) = 0.0015 mS/cm², \({g}_{\text{KL}}\) = 0.05 mS/cm², \({g}_{\text{ClL}}\) = 0.015 mS/cm², \({g}_{\text{Na}}\) = 100 mS/cm², \({g}_{\text{P}}\) = 1 mS/cm², \({g}_{\text{K}}\) = 80 mS/cm², \({g}_{\text{AHP}}\) = 1.5 mS/cm², \({E}_{\text{Ca}}\) = 120 mV, \({g}_{\text{Ca}}\) = 1 mS/cm², τ = 1000,\(\beta =4\), \({\tau }_{\text{Ca}}\) = 80 ms, \({\rho }_{\text{pump}}\) = 0.25 mM/s, \({\tau }_{\text{Ko}}\) = 2.5 s, \({\text{K}}_{\text{o}0}\) = 3.5 mM, Vol = 1.4368·10⁻⁹ cm³, \(S=4\pi {\left[3Vol/\left(4\pi \right)\right]}^{2/3}\), \({\tau }_{\text{Ki}}\) = 250 s , and \({\text{K}}_{\text{i}0}\) = 140 mM.

In the present paper, \({J}_{\text{e}}\) to modulate the firing behavior is chosen as the control parameter. In addition, \({I}_{\text{app}}\) to represent a negative square current is applied to Eq. (1), when the response of the pyramidal neuron to the inhibitory stimulation is studied.

Wang-Buzsaki model to describe the interneuron

The Wang-Buzsaki (WB) model in Ref²⁴ is used in the present paper and described as follows:

where V_i denotes the membrane voltage, \({n}_{\text{i}}\) is activated gating variable of the potassium current \({I}_{\text{Ki}}\), and \({h}_{\text{i}}\) is inactivated gating variable of the sodium current \({I}_{\text{Nai}}\). In Eqs. (9)–(11), \({C}_{\text{mi}}\) is the membrane capacitance, \({J}_{\text{i}}\) is the depolarization current, and \({I}_{\text{Li}}\) signifies the leak current. The detailed expressions of these currents are given as follows: \(I_{\text{Li}} = g_{\text{Li}} \left( {V_{\text{i}} – E_{\text{Li}} } \right)\), \(I_{\text{Ki}} = g_{\text{Ki}} n_{\text{i}}^{4} \left( {V_{\text{i}} – E_{\text{Ki}} } \right)\), and \(I_{\text{Nai}} = g_{\text{Nai}} m_{\infty \text{i}}^{3} h_{\text{i}} \left( {V_{\text{i}} – E_{\text{Nai}} } \right)\).

The equations related to the gating variables of the interneuron are as follows: \(m_{\infty i} = a_{mi} /\left( {a_{mi} + b_{mi} } \right)\), \({a}_{mi}=0.1\left({V}_{\text{i}}+35\right)/\left[1-\text{exp}\left(-\left({V}_{\text{i}}+35\right)/10\right)\right]\), \({b}_{mi}=4\text{exp}\left(-\left({V}_{\text{i}}+60\right)/18\right)\), \({a}_{ni}=0.01\left({V}_{\text{i}}+34\right)/\left[1-\text{exp}\left(-\left({V}_{\text{i}}+34\right)/10\right)\right]\), \({b}_{ni}=0.125\text{exp}\left(-\left({V}_{\text{i}}+44\right)/80\right)\), \({a}_{hi}=0.07\text{exp}\left(-\left({V}_{\text{i}}+58\right)/20\right)\), and \({b}_{hi}=1/\left[1+\text{exp}\left(-\left({V}_{\text{i}}+28\right)/10\right)\right]\).

In the present paper, \({J}_{\text{i}}\) and \({g}_{\text{Nai}}\) to modulate the spiking rate are taken as the control parameters. Other parameter values are: \({C}_{\text{mi}}\) = 1 μF/cm², \({g}_{\text{Li}}\) = 0.1 mS/cm², \({g}_{\text{Ki}}\) = 9 mS/cm², \({E}_{\text{Nai}}\) = 55 mV, \({E}_{\text{Li}}\) =  − 65 mV, and \({E}_{\text{Ki}}\) =  − 90 mV.

Coupling neuronal model and synaptic model

Inhibitory current I_GABA is introduced to Eqs. (1), (8) to form Eqs. (12), (13) (other equations remain unchanged), respectively, which are described as follows:

I_GABA modulates the electrical behavior of the membrane in Eq. (12) and \({\left[{\text{Cl}}^{-}\right]}_{\text{i}}\) in Eq. (13), since I_GABA is mediated by Cl⁻. The equation for the I_GABA is as follows:

where \({g}_{\text{GABA}}\) is the synaptic conductance, \({E}_{\text{Cl}}\) is the reversal potential, and \(s\) is the gating variable. In the present paper, \(s\) is set to 1 following a spike of the interneuron, and then obeys the following equation which is used in Ref²⁴:

with the synaptic time constant \({\tau }_{\text{GABA}}=9\) ms.

Then, a 12-dimensional coupling model containing Eqs. (12), (13), Eqs. (2), (3), (4), (5), (6), (7), Eqs. (9), (10), (11), and Eq. (15) is obtained. In the present paper, \({g}_{\text{GABA}}\) to control the inhibitory current is taken as the control parameter. The effect of the inhibitory current can be removed by setting \({g}_{\text{GABA}}=0\) mS/cm².

Network model of the pyramidal neurons

In the present paper, \(N\times N\) (\(N\) = 51) pyramidal neurons comprising a network with a square structure are considered. The lateral diffusion of potassium ion between neighboring neurons used in Refs^49,50,51 are considered as coupling. The two-dimensional pyramidal neuronal network is formulated as follows:

where the superscripts r and u represent the coordinates x and y of a neuron in the network. The lateral diffusion between neuron (r, u) and its neighboring neurons (z, w) in Eq. (20) is described as follows:

where \({\tau }_{\text{diff}}\) represents the time constant of the lateral diffusion. If the neuron (r, u) is one of the four corners of the network, it is coupled to three neighboring neurons. If the neuron (r, u) is one on the four borders exclusive the corners of the network, it is coupled to five neighboring neurons. If the neuron (r, u) is not located on the borders of the network, it is coupled to eight neighboring neurons. Then, ε^r,u,z,w = 1 for the neighboring neurons and ε^r,u,z,w = 0 for the remaining neurons. Except for \({I}_{\text{diff}}^{r,u}\), all parameters and functions of neurons in the network are the same as the isolated neurons, with \({J}_{\text{e}}\) = 4 μA. The pyramidal neurons in the network exhibit spiking when isolated. The time constant \({\tau }_{\text{diff}}\) is regarded as control parameter.

Without loss of generality, a pyramidal neuron (r = 14, u = 27) is randomly chosen as representative, which receives an inhibitory synaptic current from an inhibitory interneuron. As the inhibitory neuron exhibits enhanced activity to induce uncommon MMB in the pyramidal neuron (14, 27), the propagation of the depolarization block phase of the uncommon MMB to other neurons to form the SD in the network is studied. In the present paper, we mainly show that the uncommon MMB evoked in a pyramidal neuron can propagate in the pyramidal neuronal networks to form the SD, which are mainly influenced by the lateral diffusion of extracellular potassium ions here. In the real system, the network dynamics of SD are influenced by more factors, such as the excitatory synapses between pyramidal neurons, number of pyramidal neurons and interneurons, the topology of network composed of the pyramidal neurons and interneurons, which will be studied in future.

Calculation of the threshold for the MMB of the pyramidal neuron

The threshold for an action potential is a well-known nonlinear concept to characterize the response of a neuron to external stimulations. The threshold is an intrinsic characteristic to identify the dynamics evolving from different initial values, which can also be used to identify the dynamics induced by external stimulations, which can induce changes of the initial values. For example, there is a positive threshold for the general action potential induced from resting state by positive stimulation, and a negative threshold for the PIR spike evoked from the resting state by inhibitory stimulation^17,18,19,20. The membrane potential for the positive threshold is higher than that of the resting state, and for the negative threshold is lower than that of the resting state. The relationship between the “positive” threshold, “negative” threshold, and resting state is used as reference to understand the threshold of [K⁺]_o for the MMB.

In the present paper, we calculate the threshold for the MMB of the pyramidal neuron induced from a spiking behavior by [K⁺]_o or the inhibitory stimulations. For the equations of a neuron model at spiking behavior, an external stimulation induces the change of the variable values. At the termination time of the stimulation, the values of the variables can be regarded as the novel initial values of the model. The behaviors beginning from the novel initial values result in two behaviors, which are dependent of the stimulation. One is the MMB at first and then recovers to spiking, called suprathreshold, and the other is still spiking, called subthreshold. Then, the suprathreshold and subthreshold for the initial values in a wide range of phase space are calculated. The border between the initial values for the suprathreshold and subthreshold forms the threshold. The initial values for the suprathreshold and subthreshold comprise the MMB region and spiking region, respectively.

In the present paper, the pyramidal neuron model is eight-dimenional, then, the threshold is 8-dimensional. Projection of the threshold in two-dimensional plane or three-dimensional space, which can be visualized, is presented. [K⁺]_o is one variable of the projection plane or space, since the key dynamics for the MMB is that [K⁺]_o exhibits nearly “all-or-none” characteristic (similar to the action potential) and [K⁺]_o is the most important factor which has been related to the MMB^{21,22,24,25,26}. Then, the threshold for [K⁺]_o is mainly studied in the present paper, although other variables such as [Na⁺]_i also exhibit threshold. If the threshold for [K⁺]_o exhibits a part locating “negative” to that of the spiking, i.e., “negative” threshold, the uncommon MMB should be induced by negative stimulation. [K⁺]_o decreases at first and then runs across the “negative” threshold. If the threshold for [K⁺]_o manifests a part locates “positive” to the spiking, i.e., “positive” threshold, the excitatory stimulation or enhancement of [K⁺]_o can induce the MMB. [K⁺]_o increases at first and then passes through the “positive” threshold. Then, the threshold is used to identify the dynamics induced by [K⁺]_o pulse stimulation (Fig. 2c, d in the present paper), negative pulse stimulation (Fig. 3a, b), and negative synaptic current (Fig. 8). If the stimulation can induce spiking of pyramidal neuron run across the threshold, MMB is induced. If not, still spiking instead of MMB is induced.

Methods

The bifurcations of single pyramidal neuron are obtained using the available software XPPAUT⁵². The numerical solutions of single neuron, coupling model, and neuronal networks are solved using the fourth order Runge–Kutta method with a time step 0.001 ms.