Excitation Spreads at the Square Root of Coupling Along a Continuous Line but Stops Partway Along Weakly Coupled Nodes ── with representative values, reaching 25 m/s takes an unmyelinated diameter of 500 μm but a myelinated one of 4.166667 μm, a cross-section ratio of 14400 ── the separator is whether the path carrying excitation is continuous or discrete (whether there is myelin) ── [Paper 793]
Year: 2026
Type: preprint
Abstract: Nerve impulses travel faster in thicker fibres. It is known that fibres without myelin speed up as the square root of diameter, and myelinated fibres in proportion to diameter. Excitation along a continuous line and along discrete nodes is written with one bistable equation, and speed, and the "stopping" that occurs only in the discrete case, are counted. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──unmyelinated fibre speed scaling as the square root of diameter and myelinated fibre speed in proportion to it, the travelling-wave speed sqrt(D/2) (1-2a) of the bistable reaction--diffusion equation, and propagation failure in chains of coupled discrete cells when coupling is weak are all known (Rushton 1951, Keener 1987). The model is dimensionless──an abstract chain with the bistable reaction u(1-u)(u-a) and a=0.25; real membrane currents of nerves are not included. The link to diameter is an assumption──it concerns scaling when diffusion strength is taken proportional to diameter along a continuous line, and node spacing proportional to diameter with coupling unchanged along discrete nodes. Speed coefficients are representative──25 m/s at 500 μm unmyelinated and 6 m/s per μm myelinated are taken as order-of-magnitude values from the literature. Disease is not treated──what happens when myelin is damaged is not stated. Relation to earlier papers: Paper 272 showed that one and the same "twice as strong" opens by a factor of 6.70 on the stimulus side──its scope note stated that neural mechanisms, receptors and coding were not entered. There the correspondence between felt and stimulus magnitude was counted. Here, one step earlier, the speed at which excitation travels along a fibre is counted as set by the form of the path, and a discrete path is shown to be able to stop it. What is added is setting speeds of the two scalings side by side with representative values and deriving the cross-section ratio 14400 for 25 m/s, deriving speeds numerically in a bistable chain and showing they approach the continuous formula with strong coupling, bracketing the stopping boundary between g=0.018 and 0.019 and confirming it with a different time step and initial setup, and placing the separator on continuous versus discrete paths. First, the thickness needed for the same speed differs by orders of magnitude with the form of the path──with representative values, 25 m/s needs 500 μm unmyelinated and 4.166667 μm myelinated, a cross-section ratio of 14400 (Section 2). Second, with strong coupling, discrete nodes carry excitation as fast as a continuous line──0.996181 times the continuous formula at g=3 (Section 3). Third, with weak coupling, discrete nodes are slower than the formula──0.713506 times at g=0.05 (Section 3). Fourth, and this is the core. Along discrete nodes, weak coupling stops the excitation──at g=0.018 it does not move on to the next node even after time 3000; at g=0.019 it spreads to the end (Section 4). Fifth, the stopping is not due to the initial setup or the time step──halving the time step, or exciting 10 instead of 5 nodes at the start, g=0.018 still stops (Section 4). Sixth, the separator is whether the path carrying excitation is continuous or discrete (whether there is myelin)──"weak connections only slow things down" has no truth value until one says whether the path is continuous or discrete (Section 5). Excitation spreads at the square root of the coupling along a continuous line, but stops partway along discrete nodes when coupling is weak. With representative values, unmyelinated nerves speed up as the square root of diameter and myelinated ones in proportion, so 25 m/s needs 500 μm and 4.166667 μm, a cross-section ratio of 14400──the same speed costs different thickness by the form of the path. In a bistable chain, node speed at g=3 is 0.996181 times the continuous formula and 0.713506 times at g=0.05; at g=0.018 excitation does not move on to the next node even after time 3000, while at g=0.019 it spreads to the end, and halving the step or exciting more nodes at the start kept each case on the same side──a discrete path has a stopping boundary that a continuous path lacks. The separator is whether the path carrying excitation is continuous or discrete (whether there is myelin). Placed among the earlier papers──Paper 272 counted felt versus stimulus magnitude; this counts the speed of excitation and its stopping boundary. To be honest──the model is a dimensionless bistable chain, with neither real nerve membranes nor disease treated. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: nerve conduction velocity, myelinated nerves, saltatory conduction, bistable chain, propagation failure, reaction-diffusion. ----- 神経の興奮が伝わる速さは、線維が太いほど速い。ただし髄鞘のない線維は直径の平方根で、髄鞘のある線維は直径に比例して速くなる、と知られている。続いた線を伝わる興奮と、飛び飛びの節を伝わる興奮を一つの双安定の式で表し、速さと、飛び飛びのときだけに起きる「止まる」ことを数える。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──髄鞘のない線維の速さが直径の平方根、髄鞘のある線維の速さが直径に比例すること、双安定の反応拡散の式の進行波の速さ sqrt(D/2) (1-2a)、飛び飛びの細胞をつないだ鎖では結合が弱いと伝播が止まることは、いずれも既知である(ラシュトン 1951、キーナー 1987)。模型は無次元である──双安定の式 u(1-u)(u-a) で a=0.25 と置いた抽象的な鎖で、実在の神経の膜の電流は入れない。直径との対応は置き方である──続いた線では拡散の強さが直径に比例し、飛び飛びの節では節の間隔が直径に比例して結合が変わらない、と置いたときの尺度の話である。速さの係数は代表値である──無髄は直径 500 μm で 25 m/s、有髄は直径 1 μm あたり 6 m/s を引用の桁として置いた。病気を扱わない──髄鞘が壊れる病気で何が起きるかは述べない。既刊との関係:論文272 は、同じ「2 倍に感じる」が刺激の側では 6.70 倍ひらくことを示した──その射程注記は神経の機構を扱わず、受容器も符号化も立ち入らないと明記していた。そこでは感じ方と刺激の大きさの対応を数えた。ここではその手前、興奮が線維を伝わる速さが道の形でどう決まり、飛び飛びの道では止まりうることを数える。加えたのは、代表値で二つの尺度の速さを並べ、25 m/s に要る断面積の比 14400 を出したこと、双安定の鎖で速さを数値で出し、結合が強いと続いた線の式に近づくことを示したこと、止まる境を g=0.018 と 0.019 のあいだに挟み、刻みとはじめの置き方を変えて確かめたこと、分離子を「道が続いているか飛び飛びか」に置いたことである。 第一に、同じ速さに要る太さは、道の形で桁違いになる──代表値で 25 m/s に要る直径は無髄 500 μm、有髄 4.166667 μm、断面積の比 14400(第2節)。 第二に、結合が強いと、飛び飛びの節も続いた線と同じ速さで伝わる──結合 g=3 で続いた線の式の 0.996181 倍(第3節)。 第三に、結合が弱いと、飛び飛びの節は式より遅い──g=0.05 で 0.713506 倍(第3節)。 第四に、これが本稿の芯である。飛び飛びの節では、結合が弱いと興奮が止まる──g=0.018 では時間 3000 たっても次の節へ移らず、g=0.019 では端まで広がる(第4節)。 第五に、止まるのは、はじめの置き方や刻みのせいではない──時間の刻みを半分にしても、はじめに興奮させる節を 5 個から 10 個にしても、g=0.018 は止まる(第4節)。 第六に、分離子は、興奮を運ぶ道が続いているか飛び飛びか(髄鞘があるか)である──「つながりが弱ければ遅くなるだけ」は、道が続いているか飛び飛びかを言うまで真偽が決まらない(第5節)。 興奮は、続いた線なら結合の平方根の速さで広がり、飛び飛びの節なら結合が弱いと途中で止まる。代表値で、無髄の神経は直径の平方根、有髄の神経は直径に比例して速くなり、25 m/s に要る直径は 500 μm と 4.166667 μm、断面積は 14400 倍ちがう──同じ速さに、道の形で別の太さを払う。双安定の鎖では、結合 g=3 の節の速さは続いた線の式の 0.996181 倍、g=0.05 で 0.713506 倍に落ち、g=0.018 では時間 3000 たっても次の節へ移らず、g=0.019 では端まで広がる。刻みを半分にしても、はじめの興奮を増やしても境は同じ側にとどまった──飛び飛びの道には、続いた道に無い止まる境がある。分離子は、興奮を運ぶ道が続いているか飛び飛びか(髄鞘があるか)である。既刊との位置──論文272 は感じ方と刺激の対応を数えた。ここは興奮が伝わる速さと止まる境を数えた。正直に言えば──模型は無次元の双安定の鎖で、実在の神経の膜も病気も扱っていない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:神経の伝導速度、有髄神経、跳躍伝導、双安定の鎖、伝播の停止、反応拡散。
Author: Yuuki YAMAGISHI
Subfield: Computer Networks and Communications
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214595666
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