簇發性神經元的數學基礎 A Unified framework for Bursting Neurons
關於簇發性神經元的數學理論在來自 Aix Marseille 大學的 Saggio 教授、Spiegler 教授、Bernard 教授、Jirsa 教授於 Journal of Mathematical Neuroscience 發表論文後有了巨大的突破。他們發現了二重簡併 Taken-Bogdanov 分支 (Doubly degenerate Taken-Bogdanov bifurcation) 的展開可解釋大部份的平面簇發性神經元的產生 (planar bursters)。讓我們逐字逐句來解釋這句充滿專有名詞的句子吧!
分支 (bifurcation) 發生於微分方程因參數改變而跟著改變其行為的時候。拿 dx/dt = 1 - a x 2 為例。在這式子中,x 是狀態變數 (state variable),而 a 是參數(parameter)。當 a>0 時式子會有兩個平衡點。當 a<0 時式子沒有任何平衡點。分支發生於 a=0 時,而此時有一個平衡點。因此,分支發生於 a 由正數轉為負數的時候。這只是其中一種分支,稱為鞍點分支 (saddle-node bifurcation)。
平面簇發性神經元是簇發性神經元的一種。簇發性神經元是會在沒有外在刺激的情況下,於產生動作電位與靜止狀態間震盪的神經元 (見論文的 figure 1)。在平面簇發性神經元中,快速系統 (fast-system) 有著產生動作電位 (spiking)的機制,是二維且位於一平面上的。慢速系統 (slow-system) 則像是一個緩慢改變的參數。當慢速系統改變快速系統的『參數』時,會導致快速系統經歷分支。慢速系統可以是一維的 (遲滯現象, hysteresis) 或二維的 (慢波,slowwave)。這種週期性的經歷分支導致了這三或四維空間的簇發行為。
在2001年,Izhikevich 教授鑑定了16種可能的平面簇發性神經元。他發現其中4種分支會產生動作電位,而另外4種分支會導致產生動作電位的行為結束,因此共有16種可能的平面簇發性神經元。本篇論文更近一步地把它們分為4種亞型(subtypes),因此共有64種不同的簇發性神經元亞型。
展開 (unfolding) 是指快速系統改變慢速系統的參數。前面提到分支發生於快速系統的參數改變時。數學上,我們分析快速系統時會把慢速系統當作是停滯在同一時間點上,因此我們可以把慢速系統當作快速系統的『參數』。有時候這變化很微小,有時候則很極端,如平衡點的消失或週期性解的出現。這些突然的、定性的變化即為分支。
二重簡併 Taken-Bogdanov 分支是一種簡併分支(degenerate bifurcation),因兩個發現它的數學家而得名。以一個十分簡化的方式來說,它可以被想成是三個分支同時發生的情況。因為有三個分支,所以有三個參數。二重簡併是指系統經歷兩次失敗的簡併,即所有混合二階導數 (mixed second derivative) 在定點上為零。這種分支有趣的點在於其他正常的分支,如鞍點分支或霍普夫分岔(Hopf bifurcations),存在於它的周圍。因此,這種分支附近有著所有簇發性神經元所需的元素。
作者們重建了所有16個平面簇發性神經元,但只重建了64個亞型中的40個 (見figure 13)。他們以二重簡併 Taken-Bogdanov 的算式為快速系統,而慢速系統則展開球體表面的三個參數 (見 figure 3)。這裡的重點在於簇發性神經元從來不會自己通過二重簡併 Taken-Bogdanov 的點。整個系統只會在它周圍移動。
因此作者們選擇了以二重簡併 Taken-Bogdanov 為中心的球體表面。當快速系統的參數被限制於表面時,快速系統可用可被控制的方式跨越各種不同分支。這樣的模型給予幾乎所有平面簇發性神經元的亞型一個統一的解釋,也因此為神經科學奠定研究簇發性神經元的基礎。
The mathematical theory behind bursting neurons got a big boost when professors, Dr. Maria Luisa Saggio, Dr. Andreas Spiegler, Dr. Christophe Bernard,. and Dr. Viktor K. Jirsa, from Aix Marseille University published their paper in the Journal of Mathematical Neuroscience. They found an unfolding of a doubly degenerate Taken-Bogdanov bifurcation capable of producing a majority of all possible planar bursters. That is one jargoned filled sentence; let us take it one word at a time.
A bifurcation occurs when a differential equation changes its behavior as a parameter changes. For example, take the equation dx/dt = 1 – a x2. Here, x is our state variable, and a is our parameter. When a>0 there are two equilibrium points. When a<0 there are no equilibrium points. The bifurcation occurs when a=0 and there is a single equilibrium point. Thus, the bifurcation occurs as a changes from positive to negative. This bifurcation is one of many and is known as the saddle-node bifurcation.
A planar burster is a class of bursting neurons. Recall a bursting neuron is any neuron that oscillates between a spiking state and rest state without any external input (see figure 1 in the paper). In a planar burster, the fast-system contains the spiking mechanism, and is only two-dimensional and lies on a plane. The slow-system acts like a slowly changing parameter. As the slow-system changes the “parameters” of the fast-system this causes the fast-system to undergo bifurcations. The slow-system can be either one-dimensional (hysteresis), or two-dimensional (slow wave). The periodic occurrence of bifurcations gives rise to the bursting behavior of the whole three or four-dimensional system.
In 2001, Dr. Izhikevich was able to identify all 16 possible Planar bursters. He realized that there are 4 possible bifurcations responsible for spiking to start, and four possible bifurcations for spiking to end, thus 16 different choices for planar bursters. This paper further subdivides them into 4 subtypes for a total of 64 different burster subtypes. Unfolding refers to the slow-system changing the parameters of the fast-system. Recall a bifurcation occurs when the parameters of the fast-system change. Mathematically we analyze the fast-system as if the slow-system was fixed in time, and thus, we can think of the slow variables as the “parameters” of the fast-system. Sometimes the changes are subtle, sometimes they are extreme, like an equilibrium point disappearing, or a periodic solution appearing. These sudden qualitative changes are the bifurcations.
A doubly degenerate Taken-Bogdanov is a type of degenerate bifurcation named after the two mathematicians that discovered it. In an oversimplified sense, it can be thought of as three bifurcations occurring at the same time. Because there are three different bifurcations, there are 3 parameters. The double degeneracy condition refers to the fact that the system fails the degeneracy condition twice, i.e. all mixed second derivative are zero at the fixed point. The interesting thing about this bifurcation is that all other regular bifurcations, like Saddle Nodes, and Hopf Bifurcations, exist around it. Thus, this bifurcation has all the ingredients for bursters nearby.
The authors are able to recreate all 16 planar bursters, but only 40 of the 64subtypes (see figure 13). They use the equation for the doubly degenerate Taken-Bogdanov for the fast-system and the slow-system unfolds the 3 parameters on the surface of a sphere (see figure 3). The key point is a bursting neuron never passes through the doubly degenerate Taken-Bogdanov point itself. The full system only moves around it.
Thus, the authors chose the surface of a sphere, with the doubly degenerate Taken-Bogdanov point at the center. When the fast-system’s parameters are constrained to the surface of the sphere the fast-system can cross through all the different possible bifurcations in a controlled manner. This framework provides a unified description for almost all subtypes of planar bursters, and thus gives neuroscience a mathematical foundation to study actual bursting neurons.
Source: https://mathematical-neuroscience.springeropen.com/articles/
10.1186/s13408-017-0050-8
撰稿:Alex White
翻譯:劉沛弦
分支 (bifurcation) 發生於微分方程因參數改變而跟著改變其行為的時候。拿 dx/dt = 1 - a x 2 為例。在這式子中,x 是狀態變數 (state variable),而 a 是參數(parameter)。當 a>0 時式子會有兩個平衡點。當 a<0 時式子沒有任何平衡點。分支發生於 a=0 時,而此時有一個平衡點。因此,分支發生於 a 由正數轉為負數的時候。這只是其中一種分支,稱為鞍點分支 (saddle-node bifurcation)。
平面簇發性神經元是簇發性神經元的一種。簇發性神經元是會在沒有外在刺激的情況下,於產生動作電位與靜止狀態間震盪的神經元 (見論文的 figure 1)。在平面簇發性神經元中,快速系統 (fast-system) 有著產生動作電位 (spiking)的機制,是二維且位於一平面上的。慢速系統 (slow-system) 則像是一個緩慢改變的參數。當慢速系統改變快速系統的『參數』時,會導致快速系統經歷分支。慢速系統可以是一維的 (遲滯現象, hysteresis) 或二維的 (慢波,slowwave)。這種週期性的經歷分支導致了這三或四維空間的簇發行為。
在2001年,Izhikevich 教授鑑定了16種可能的平面簇發性神經元。他發現其中4種分支會產生動作電位,而另外4種分支會導致產生動作電位的行為結束,因此共有16種可能的平面簇發性神經元。本篇論文更近一步地把它們分為4種亞型(subtypes),因此共有64種不同的簇發性神經元亞型。
展開 (unfolding) 是指快速系統改變慢速系統的參數。前面提到分支發生於快速系統的參數改變時。數學上,我們分析快速系統時會把慢速系統當作是停滯在同一時間點上,因此我們可以把慢速系統當作快速系統的『參數』。有時候這變化很微小,有時候則很極端,如平衡點的消失或週期性解的出現。這些突然的、定性的變化即為分支。
二重簡併 Taken-Bogdanov 分支是一種簡併分支(degenerate bifurcation),因兩個發現它的數學家而得名。以一個十分簡化的方式來說,它可以被想成是三個分支同時發生的情況。因為有三個分支,所以有三個參數。二重簡併是指系統經歷兩次失敗的簡併,即所有混合二階導數 (mixed second derivative) 在定點上為零。這種分支有趣的點在於其他正常的分支,如鞍點分支或霍普夫分岔(Hopf bifurcations),存在於它的周圍。因此,這種分支附近有著所有簇發性神經元所需的元素。
作者們重建了所有16個平面簇發性神經元,但只重建了64個亞型中的40個 (見figure 13)。他們以二重簡併 Taken-Bogdanov 的算式為快速系統,而慢速系統則展開球體表面的三個參數 (見 figure 3)。這裡的重點在於簇發性神經元從來不會自己通過二重簡併 Taken-Bogdanov 的點。整個系統只會在它周圍移動。
因此作者們選擇了以二重簡併 Taken-Bogdanov 為中心的球體表面。當快速系統的參數被限制於表面時,快速系統可用可被控制的方式跨越各種不同分支。這樣的模型給予幾乎所有平面簇發性神經元的亞型一個統一的解釋,也因此為神經科學奠定研究簇發性神經元的基礎。
The mathematical theory behind bursting neurons got a big boost when professors, Dr. Maria Luisa Saggio, Dr. Andreas Spiegler, Dr. Christophe Bernard,. and Dr. Viktor K. Jirsa, from Aix Marseille University published their paper in the Journal of Mathematical Neuroscience. They found an unfolding of a doubly degenerate Taken-Bogdanov bifurcation capable of producing a majority of all possible planar bursters. That is one jargoned filled sentence; let us take it one word at a time.
A bifurcation occurs when a differential equation changes its behavior as a parameter changes. For example, take the equation dx/dt = 1 – a x2. Here, x is our state variable, and a is our parameter. When a>0 there are two equilibrium points. When a<0 there are no equilibrium points. The bifurcation occurs when a=0 and there is a single equilibrium point. Thus, the bifurcation occurs as a changes from positive to negative. This bifurcation is one of many and is known as the saddle-node bifurcation.
A planar burster is a class of bursting neurons. Recall a bursting neuron is any neuron that oscillates between a spiking state and rest state without any external input (see figure 1 in the paper). In a planar burster, the fast-system contains the spiking mechanism, and is only two-dimensional and lies on a plane. The slow-system acts like a slowly changing parameter. As the slow-system changes the “parameters” of the fast-system this causes the fast-system to undergo bifurcations. The slow-system can be either one-dimensional (hysteresis), or two-dimensional (slow wave). The periodic occurrence of bifurcations gives rise to the bursting behavior of the whole three or four-dimensional system.
In 2001, Dr. Izhikevich was able to identify all 16 possible Planar bursters. He realized that there are 4 possible bifurcations responsible for spiking to start, and four possible bifurcations for spiking to end, thus 16 different choices for planar bursters. This paper further subdivides them into 4 subtypes for a total of 64 different burster subtypes. Unfolding refers to the slow-system changing the parameters of the fast-system. Recall a bifurcation occurs when the parameters of the fast-system change. Mathematically we analyze the fast-system as if the slow-system was fixed in time, and thus, we can think of the slow variables as the “parameters” of the fast-system. Sometimes the changes are subtle, sometimes they are extreme, like an equilibrium point disappearing, or a periodic solution appearing. These sudden qualitative changes are the bifurcations.
A doubly degenerate Taken-Bogdanov is a type of degenerate bifurcation named after the two mathematicians that discovered it. In an oversimplified sense, it can be thought of as three bifurcations occurring at the same time. Because there are three different bifurcations, there are 3 parameters. The double degeneracy condition refers to the fact that the system fails the degeneracy condition twice, i.e. all mixed second derivative are zero at the fixed point. The interesting thing about this bifurcation is that all other regular bifurcations, like Saddle Nodes, and Hopf Bifurcations, exist around it. Thus, this bifurcation has all the ingredients for bursters nearby.
The authors are able to recreate all 16 planar bursters, but only 40 of the 64subtypes (see figure 13). They use the equation for the doubly degenerate Taken-Bogdanov for the fast-system and the slow-system unfolds the 3 parameters on the surface of a sphere (see figure 3). The key point is a bursting neuron never passes through the doubly degenerate Taken-Bogdanov point itself. The full system only moves around it.
Thus, the authors chose the surface of a sphere, with the doubly degenerate Taken-Bogdanov point at the center. When the fast-system’s parameters are constrained to the surface of the sphere the fast-system can cross through all the different possible bifurcations in a controlled manner. This framework provides a unified description for almost all subtypes of planar bursters, and thus gives neuroscience a mathematical foundation to study actual bursting neurons.
Source: https://mathematical-neuroscience.springeropen.com/articles/
10.1186/s13408-017-0050-8
撰稿:Alex White
翻譯:劉沛弦
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