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The Bifurcation and Multi-timescale Singularity Analysis of the AII Amacrine Cell Firing Activities in Retina

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Abstract

A well-established AII amacrine cell model linked to blinding retinal diseases demonstrates rhythmic bursting activity. This behavior potentially impedes the retinal and cortical responses essential for visual restoration. Consequently, understanding the dynamics of AII amacrine cell firing is critical. We expanded the study by incorporating a calcium ion channel into the original model, creating a refined eight-dimensional AII model. This enhancement allowed us to conduct an in-depth exploration of system behavior via bifurcation analysis and an extensive examination of multi-timescale singularities. We identified the parameter ranges that control the firing activities of the model through codimension-one and codimension-two bifurcation analyses of key parameters. Furthermore, our research revealed that introducing a balance equation determines a transition from regular spiking to bursting. Importantly, the orchestration of three-timescale spiking and bursting attractors is enabled by the universal unfolding of the winged cusp singularity. This study provides a comprehensive analysis of firing dynamics in a high-dimensional neuron model. The insights derived significantly advance our understanding the dynamical behaviors of AII amacrine cells and are promising for improving strategies for retinal pathologies.

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Acknowledgements

The authors express their sincere gratitude to the anonymous referee for their careful review and invaluable suggestions, which have greatly enhanced the quality of this paper. They also acknowledge the partial support from the National Natural Science Foundation of China (Grant Nos. 11872183) and the China Scholarship Council (Grant Nos. 202306150098).

Author information

Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Na Zhao, Jian Song, Ke He & Shenquan Liu

  2. School of Mathematical and Computational Sciences, Massey University, Auckland, 4442, New Zealand

    Jian Song

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  1. Na Zhao
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  2. Jian Song
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Contributions

Zhao was involved in conceptualization, methodology, and software, and wrote the main manuscript text. Song and He were responsible for software and data curation, and reviewed the manuscript. Liu reviewed and submitted the manuscript.

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Correspondence to Shenquan Liu.

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Appendices

Model Parameter Description

See Tables 1 and 2.

Table 1 Corresponding definitions and default values of parameters in the AII model
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Table 2 Default values of the parameters in steady-state function
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Correlation Verification of the Bifurcation Points

1.1 Hopf Bifurcation Points

The firing dynamics of system (1) are predominantly impacted by the Hopf bifurcation points situated in both the upper and lower branches of the bifurcation curve. We will focus on Hopf bifurcation point \(H_{1}\) to analyze the criticality of the point. To commence this analysis, we first reformulate system (1) as follows:

$$\begin{aligned} \begin{aligned} \dot{V}&=F_{1}(V,m_{\text {Na}}, h_{\text {Na}}, m_{\text {Ca}},m_{M}, m_{A},h_{A}^{1},h_{A}^{2}),\\ \dot{x}&=F_{i}(V,x), \quad x = m_{\text {Na}},h_{\text {Na}}, m_{\text {Ca}},m_{M},m_{A},h_{A}^{1},h_{A}^{2}\quad i=2,\ldots ,8,\\ \end{aligned} \end{aligned}$$
(18)

where

$$\begin{aligned} F_{1}=&\frac{1}{C_{m}}[I_{\text {inject}}-g_{\text {Na}}m_{\text {Na}}^{3}h_{\text {Na}} (V-E_{\text {Na}})-g_{\text {Ca}}m_{\text {Ca}} (V-E_{\text {Ca}})-g_{M}\\&m_{M} (V-E_{K})-g_{A}m_{A}[ch_{A}^{1}+ (1-c)h_{A}^{2}] (V-E_{K})-g_{L} (V-E_{L})],\\ F_{i}=&\frac{x_{\infty }-x}{\tau _{x}}, \quad x = m_{\text {Na}},h_{\text {Na}}, m_{\text {Ca}},m_{M},m_{A},\quad i=2,\ldots ,6,\\ F_{7}=&\frac{h_{A,\infty }-h_{A}^{1}}{\tau _{h_{A}^{1}}},\\ F_{8}=&\frac{h_{A,\infty }-h_{A}^{2}}{\tau _{h_{A}^{2}}}.\\ \end{aligned}$$

The Jacobian matrix of the above equation is

$$\begin{aligned} \begin{aligned} A&=\left( \begin{array}{llllllll} \frac{\partial F_{1}}{\partial V} &{} \frac{\partial F_{1}}{\partial m_{\text {Na}}} &{} \frac{\partial F_{1}}{\partial h_{\text {Na}}} &{} \frac{\partial F_{1}}{\partial m_{\text {Ca}}} &{} \frac{\partial F_{1}}{\partial m_{M}} &{} \frac{\partial F_{1}}{\partial m_{A}} &{} \frac{\partial F_{1}}{\partial h_{A}^{1}} &{} \frac{\partial F_{1}}{\partial h_{A}^{2}}\\ \frac{\partial F_{2}}{\partial V} &{} -100 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{\partial F_{3}}{\partial V} &{} 0 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{\partial F_{4}}{\partial V} &{} 0 &{} 0 &{} \frac{\partial F_{4}}{\partial m_{\text {Ca}}} &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{\partial F_{5}}{\partial V} &{} 0 &{} 0 &{} 0 &{} -1/50 &{} 0 &{} 0 &{} 0\\ \frac{\partial F_{6}}{\partial V} &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0\\ \frac{\partial F_{7}}{\partial V} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{\partial F_{7}}{\partial h_{A}^{1}} &{} 0\\ \frac{\partial F_{8}}{\partial V} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{\partial F_{8}}{\partial h_{A}^{2}}\\ \end{array} \right) \\ \end{aligned} \end{aligned}$$
(19)

Given the intricacies of the model (1), we intentionally refrain from presenting comprehensive expressions to describe it in detail. When

\(g_{M}=g_{M}^{H_{1}}=0.01181~S/\) \(cm^{2}\), we have calculated the equilibrium of system (1) as \((V^{*},m_{\text {Na}}^{*},h_{\text {Na}}^{*},m_{\text {Ca}}^{*}\), \(m_{M}^{*},m_{A}^{*}, h_{A}^{1,*},h_{A}^{2,*})\)

\(=(-40.30491,0.82333,0.00126,0.99261,0.48101,0.01306,0.56482,0.56482)\). Concurrently, the Jacobian matrix at \(H_{1}\) is

$$\begin{aligned} \begin{aligned} \left( \begin{array}{llllllll} -8.396 &{} 366.405 &{} 10079.8 &{} 130.305 &{} -433.185 &{} -1657.96 &{} -22.053 &{} -16.126\\ 2.909 &{} -100 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ -0.009 &{} 0 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0.0001 &{} 0 &{} 0 &{} -0.056 &{} 0 &{} 0 &{} 0 &{} 0\\ 0.001 &{} 0 &{} 0 &{} 0 &{} -1/50 &{} 0 &{} 0 &{} 0\\ 0.002 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0\\ -0.005 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -0.052 &{} 0\\ -0.001 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -0.01\\ \end{array} \right) \\ \end{aligned}\nonumber \\ \end{aligned}$$
(20)

Furthermore, by calculating the eigenvalues, we obtain a pair of complex conjugate eigenvalues \(\lambda \) and \(\bar{\lambda }\), where \(\lambda =i\omega \) and \(\omega =9.44707\). This result confirms that system (1) undergoes a Hopf bifurcation at \(H_{1}\). Subsequently, we employ the MatCont software package to compute the first Lyapunov coefficient, yielding a value of \(0.20826>0\). Thus, \(H_{1}\) represents a subcritical Hopf bifurcation point that generates an unstable limit cycle.

Similar analytical procedures can be applied to the other bifurcation points, as detailed in Table 3.

Table 3 Data of the codimension-one bifurcation
Full size table

1.2 Generalized Hopf (Bautin) Points

Regarding the codimension-two bifurcation points, it is essential to highlight that generalized Hopf (Bautin) points, are the distinctive positions that satisfy specific conditions while ensuring the existence of a pair of purely imaginary eigenvalues. Detailed definitions and explanations of these points can be found in established literature (Kuznetsov et al. 1998). We calculate the bifurcation diagram using the Matcont software package. The relevant data of generalized Hopf (Bautin) points in the codimension-two bifurcation Fig. 8 are shown in Table 4.

Table 4 Data of the codimension-two bifurcation in \((g_{M},g_{\text {Ca}})\) plane
Full size table

In the neighbor of Bautin bifurcation points \(GH_{i}(i=1,2)\), the local topology of the differential system (1) is equivalent to the following normalization form:

$$\begin{aligned} \begin{aligned} \dot{z}=&(\beta _{1}+i)z+\beta _{2}z|z|^{2}+sz|z|^{4}, \quad z\in \mathbb {C}^{1};\\ \dot{\xi }_{-}=&-\xi _{-}, \quad \xi _{-}\in \mathbb {R}^{2}, \end{aligned} \end{aligned}$$
(21)

where \(\beta _{1},\beta _{2}\in \mathbb {R}, s=\hbox {sign}(l_{2}))=+1\).

1.3 Transcritical Bifurcation Point

The fixed equation of system (3) satisfies

$$\begin{aligned} C_{m}\dot{V}&=I_{\text {inject}} - I_{\text {Na}} - I_{\text {Ca}} - I_{M} - I^{*}_{A} - I_{L}=0 ,\\ \dot{x}&=\frac{x_{\infty }-x}{\tau _{x}}=0, x = m_{\text {Na}}, h_{\text {Na}}, m_{\text {Ca}},m_{M}, m_{A},{h}_{A}^{1},\\ \end{aligned}$$

Simple calculation can obtain \(x=x_{\infty }(V), x = m_{\text {Na}}, h_{\text {Na}}, m_{\text {Ca}},m_{M}, m_{A},{h}_{A}^{1}\). Letting

$$\begin{aligned} \sum _{i} g_{i}m_{i,\infty }^{ai}h_{i,\infty }^{bi} (V-E_{i})&= g_{\text {Na}}m_{\text {Na},\infty }^{3}h_{\text {Na},\infty } (V-E_{\text {Na}})\\&\quad +g_{\text {Ca}}m_{\text {Ca},\infty } (V-E_{\text {Ca}})+g_{M}m_{M,\infty } (V-E_{K})\\&\quad +g_{A}m_{A,\infty }[ch_{A,\infty }^{1}+ (1-c)\bar{h}_{A}^{2}] (V-E_{K})\\&\quad +g_{L} (V-E_{L}), \end{aligned}$$

The associated resulting fixed point equation is

$$\begin{aligned} \begin{aligned} I_{\text {inject}} - \sum _{i} g_{i}m_{i,\infty }^{a_{i}}h_{i,\infty }^{b_{i}} (V-E_{i})=0, \end{aligned} \end{aligned}$$
(22)

Given the bifurcation parameter \(g_{\text {Ca}}\in \{g_{i},E_{i}\}_{i\in I}\), there are exists a solution \( (V,g_{\text {Ca}})= (V_{c},g_{\text {Cac}})\) satisfying the two degeneracy conditions (5) and (6). Then solve the fixed point equation (22), the following result can be obtained:

$$\begin{aligned} I_{\text {inject}}^{c}=&\sum _{i} g_{i}m_{i,\infty }^{a_{i}} (V_{c})h_{i,\infty }^{b_{i}} (V_{c}) ( (V_{c})-E_{i}), \end{aligned}$$

and the transcritical bifurcation occurs at this critical condition in system (1) (Franci et al. 2012, 2013).

Let \(V_{1}=V+V_{c}, x_{c}=-(x_{\infty }-x)\), and continue to denote \(V_{1}, x_{c}\) imply as V, x. Then, we linearize system (3) in the equilibrium \(V_{c},x_{\infty }\). The center space \(E_{c}\) associated with the degeneracy conditions (5) and (6) is spanned by the vector

$$\begin{aligned} \vec {v}_{c}= (1,\vec {k}), \qquad \quad ~ \vec {k}:=[k_{x}], \qquad \quad ~ k_{x}:=\frac{\partial x_{\infty }}{\partial V}\bigg |_{V=V_{c}}, \end{aligned}$$

where k is the vector whose elements are the slopes of the (in)activation functions of the gating variables calculated at \(V=V_{c}\). The associated center manifold \(\mathbb {M}_{c}\) is exponentially attractive. Because the remaining nonzero eigenvalues of the Jacobian (4) are all negative when (5) and (6) are satisfied.

For the first order in \(V_{1}-V_{c}\), the dynamics on \(\mathbb {M}_{c}\) is given by:

$$\begin{aligned} \begin{aligned} C_{m}\dot{V_{1}}=- I_{\text {Na}}^{\star } - I_{\text {Ca}}^{\star } - I_{M}^{\star } - I^{\star }_{A} - I_{L} -I_{\text {inject}}^{c}+I_{\text {inject}} =0,\\ \end{aligned} \end{aligned}$$
(23)

where

$$\begin{aligned} I_{\text {Na}}^{\star }&=g_{\text {Na}} (m_{\text {Na},\infty } (V_{c})+k_{m_{\text {Na}}} (V_{1}-V_{c}))^{3} (h_{\text {Na},\infty } (V_{c})+k_{h_{\text {Na}}} (V_{1}-V_{c})) (V_{1}-E_{\text {Na}}),\\ I_{\text {Ca}}^{\star }&=g_{\text {Ca}} (m_{\text {Ca},\infty } (V_{c})+k_{m_{\text {Ca}}} (V_{1}-V_{c})) (V_{1}-E_{\text {Ca}}),\\ I_{M}^{\star }&=g_{M} (m_{M,\infty } (V_{c})+k_{m_{M}} (V_{1}-V_{c})) (V_{1}-E_{K}), \\ I_{A}^{\star }&=g_{A} (m_{A,\infty } (V_{c})+k_{m_{A}} (V_{1}-V_{c}))[c (h_{A,\infty }^{1} (V_{c})+k_{h_{A}^{1}} (V_{1}-V_{c}))+ (1-c)\bar{h}_{A}^{2}]\\&~~~ (V_{1}-E_{K}).\\ \end{aligned}$$

Through affine reparameterization, the stimulus current was varied as follows:

$$\begin{aligned} \bar{g}_{\text {Ca}}&=g_{\text {Ca}},\\ \bar{I}_{\text {inject}}&=I_{\text {inject}}^{c}-\frac{\partial \dot{V}}{\partial g_{\text {Ca}}}\bigg |_{\begin{array}{c} g_{\text {Ca}}=g_{\text {Cac}}\\ V_{1}=V_{c} \end{array}} (g_{\text {Ca}}-g_{\text {Cac}}). \end{aligned}$$

Then, we verified that the center manifold dynamics satisfy

$$\begin{aligned} \dot{V_{1}}\bigg |_{\begin{array}{c} g_{\text {Ca}}=g_{\text {Cac}}\\ V_{1}=V_{c} \end{array}}=\frac{\partial \dot{V_{1}}}{\partial V_{1}}\bigg |_{\begin{array}{c} g_{\text {Ca}}=g_{\text {Cac}}\\ V_{1}=V_{c} \end{array}}=\frac{\partial \dot{V_{1}}}{\partial \bar{g}_{\text {Ca}}}\bigg |_{\begin{array}{c} g_{\text {Ca}}=g_{\text {Cac}}\\ V_{1}=V_{c} \end{array}}=0, \end{aligned}$$

which corresponds to the defining condition of a transcritical bifurcation (Seydel 2009).

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Zhao, N., Song, J., He, K. et al. The Bifurcation and Multi-timescale Singularity Analysis of the AII Amacrine Cell Firing Activities in Retina. J Nonlinear Sci 34, 93 (2024). https://doi.org/10.1007/s00332-024-10074-y

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