Gary's LIF Ordering SystemGary's LIF Ordering System

§1Definition: what is actually different in this LIF system?定義：這個 LIF 系統真正不同的是什麼？

For M6 the smallest formally validated variants use only six adjustable values per run, while the Navigator has five dimensions. No 720-way classifier or 720-entry lookup is required.

M6 目前最小且正式通過的版本，每次 run 只需要 6 個可調數字，而 Navigator 只有 5 維；沒有 720-way classifier，也沒有 720-entry lookup。

§2The continuous LIF equation used in 3.0–3.23.0–3.2 使用的 continuous LIF 方程

For the clean first-spike experiments, set \(V_i(0)=0\), take \(V_{rest}=0\), and hold \(I_i\) constant before the first spike.

乾淨的 first-spike 實驗先令 \(V_i(0)=0\)、\(V_{rest}=0\)，並在 first spike 前令 \(I_i\) 為常數。

Because the representation uses only the first spike, post-spike reset is irrelevant to the 3.0–3.2 ordering definition.

因為 representation 只讀第一個 spike，所以 3.0–3.2 的 ordering 定義不依賴 first spike 後的 reset。

§3Every variable in the function函式裡每一個東西的意思

§4Exact first-spike solution and exact gradientsExact first-spike 解與 exact gradients

This is why 3.0–3.2 can differentiate first-spike time directly instead of differentiating a hard binary spike.

這就是 3.0–3.2 可以直接對 first-spike time 微分，而不需要對 hard binary spike 微分的原因。

Concrete point: with \(I=2,\tau=1,V_{th}=1,R=1\), \(t^*=0.6931\), while the derivatives are \(-0.5,\ 0.6931,\ 1,\ -1\) respectively.

具體數字：若 \(I=2,\tau=1,V_{th}=1,R=1\)，則 \(t^*=0.6931\)，四個 derivative 分別是 \(-0.5,\ 0.6931,\ 1,\ -1\)。

§5Factorial first-spike ordering spaceFactorial first-spike ordering space

The state is relational: no single neuron “is” the class. A state is the complete relative temporal order among all neurons.

state 是 relational：不是某一顆 neuron 代表 class，而是全部 neurons 的相對 first-spike 次序共同代表一個 state。

§6The 5-D Navigator5-D Navigator

We call the target-adjacent first-spike gaps the Navigator. For M6 it has exactly five dimensions.

我們把 target order 裡相鄰的 first-spike gaps 正式叫 Navigator。M6 剛好只有五維。

Example target \(\pi^*=(2,3,1,4,5,6)\) requires \(t_2

例如 target \(\pi^*=(2,3,1,4,5,6)\)，要求 \(t_2

§7Navigator lossNavigator loss

The gradient is negative, so reducing loss means increasing every target-adjacent gap \(g_k\). If a pair is reversed, the loss produces a strong correction; if it is correct but too close to the boundary, it still receives a margin correction.

這個 gradient 永遠是負的，所以降低 loss 就是把 target-adjacent gap \(g_k\) 推大。pair 如果反了，修正很強；pair 雖然順序對但太靠近 boundary，仍會收到 margin correction。

§8Full gradient path: target → physical LIF parameter完整梯度鏈：target → physical LIF parameter

For a target-adjacent pair, \(g_k=t_b-t_a\). Therefore \(\partial g_k/\partial t_a=-1\) and \(\partial g_k/\partial t_b=+1\). A violated target relation directly pushes the earlier neuron earlier and/or the later neuron later.

對 target-adjacent pair，\(g_k=t_b-t_a\)，所以 \(\partial g_k/\partial t_a=-1\)、\(\partial g_k/\partial t_b=+1\)。如果 target relation 錯了，gradient 會直接把該早的 neuron 推早、該晚的 neuron 推晚。

Physical values can be bounded by an unconstrained variable \(a_i\):

physical value 可用 unconstrained variable \(a_i\) 做 bounded map：

§9What is actually learnable?到底什麼是 learnable？

“Learnable LIF parameter” means that the gradient directly updates a physical neuron parameter such as \(\tau_i,V_{th,i},R_i\). It does not yet mean that a reusable train-once addressing function has been learned.

「LIF parameter 可學」的精確意思是：gradient 直接更新 \(\tau_i,V_{th,i},R_i\) 這類 physical neuron parameter。它還不等於已經學會一個 train-once、可重複使用的 addressing function。

§10Experiment 3.0 — Direct Navigator addressingExperiment 3.0 — Navigator 直接 addressing

Question: can a frozen bounded-current LIF be pushed into an arbitrary target permutation using only the 5-D Navigator and gradient descent?

問題：固定 bounded-current LIF，只給 target permutation 與 5-D Navigator，能不能用 gradient descent 直接推進任意 target chamber？

3.0.1 covers all 720 target permutations with 10 initializations per target. All 7,200 runs pass both hard ordering and the strict \(\delta=0.1\) margin.

3.0.1 覆蓋全部 720 targets，每個 target 10 個 initializations；7,200/7,200 都通過 hard ordering 與 strict \(\delta=0.1\) margin。

§11Experiment 3.1 — Direct state-to-state navigationExperiment 3.1 — 直接 state-to-state navigation

3.1 starts from a valid canonical source state and changes the target without resetting to a random initialization. All 518,400 ordered source→target transitions pass hard ordering and margin.

3.1 從合法 canonical source state 出發，不 reset 成 random initialization，直接換 target；518,400 個 ordered source→target transitions 全部通過 hard ordering 與 margin。

This is exact reachability without a supplied symbolic swap path, but not a shortest-path theorem. Only 291,039 / 517,680 nonidentity transitions (56.2199%) had sampled path length equal to Kendall shortest distance; 43.7801% had extra flips. Mean efficiency was 0.887137 and the minimum was 1/3.

這證明「不用 supplied symbolic swap path 也能 exact reach target」，但不是 shortest-path theorem。517,680 個 nonidentity transitions 中只有 291,039（56.2199%）的 sampled path length 等於 Kendall 最短距離；43.7801% 有額外 flips。mean efficiency=0.887137，最低=1/3。

The sampled Kendall distance was non-increasing at every sampled step in 0.586572 of transitions, so gradient navigation is exact but often non-monotone in hard permutation space.

sampled Kendall distance 每一步都不增加的比例只有 0.586572，所以 gradient navigation 最終 exact，但在 hard permutation space 裡常會繞路。

§12Experiment 3.2 — Intrinsic parameter controllabilityExperiment 3.2 — intrinsic parameter controllability

The Navigator and loss are unchanged. Only the physical parameter receiving the gradient is changed.

Navigator 與 loss 完全不變，只改「gradient 最後更新哪一個 physical parameter family」。

Each arm is 720 targets × 5 initializations = 3,600 runs. All four arms pass every target/initialization combination.

每個 arm 都是 720 targets × 5 initializations = 3,600 runs；四個 arms 全部通過。

§13Experiment 3.3 — Signed voltage-mediated couplingExperiment 3.3 — signed voltage-mediated coupling

3.3 freezes current and intrinsic parameters and learns only cross-node coupling:

3.3 固定 current 與 intrinsic parameters，只學 cross-node coupling：

Formal settings: \(dt=0.02\), horizon \(=2.5\), \(|W_{ij}|<0.08\).

formal 設定：\(dt=0.02\)、horizon \(=2.5\)、\(|W_{ij}|<0.08\)。

The evidence therefore supports rank-coded net incoming balance, not yet source-specific topology discovery. The 30-edge arm does not outperform the six-value row-shared arm in optimization speed. Across final directed edges, the reported sign fractions were about 0.32647 positive, 0.67352 negative, and 0.000015 near zero.

所以目前 evidence 支持的是 rank-coded net incoming balance，不是 source-specific topology discovery；30-edge arm 沒有比 6-value row-shared arm 更快。directed arm 的 final edge sign 比例約為 positive 0.32647、negative 0.67352、near-zero 0.000015。

§14What snnTorch snn.Leaky actually implementssnnTorch 的 snn.Leaky 到底在做什麼

The official snnTorch 1.0.0 documentation describes snn.Leaky as a discrete-time first-order LIF. With reset-by-subtraction:

官方 snnTorch 1.0.0 文件把 snn.Leaky 定義成 discrete-time first-order LIF。reset-by-subtraction 時：

Each call advances one timestep, so a normal forward simulation explicitly loops over time. The neuron returns spike and membrane state. The API supports learn_beta=True and learn_threshold=True.

每 call 一次只前進一個 timestep，所以一般 forward 會明確 loop over time；neuron 回傳 spike 與 membrane state。API 支援 learn_beta=True 與 learn_threshold=True。

Therefore “learnable decay” or “learnable threshold” alone is not the novelty of Gary's system; snnTorch already supports those ideas.

所以「decay 可學」或「threshold 可學」本身不是 Gary's system 的 novelty；snnTorch 已經支援。

§15Deep comparison: Gary's system vs snnTorch Leaky深度比較：Gary's system vs snnTorch Leaky

A conceptual relation between continuous time constant and discrete decay is \(\beta\approx e^{-\Delta t/\tau}\), but the exact physical mapping depends on discretization and input scaling, so they should not be treated as identical parameterizations.

continuous time constant 和 discrete decay 的概念關係常寫成 \(\beta\approx e^{-\Delta t/\tau}\)，但 exact physical mapping 取決於 discretization 與 input scaling，所以不能把兩者當逐項完全一樣。

§16snnTorch RLeaky vs Experiment 3.3snnTorch RLeaky vs Experiment 3.3

The official snn.RLeaky recurrence adds a recurrent function of output spikes:

官方 snn.RLeaky recurrence 加的是 output spikes 的 recurrent function：

Experiment 3.3 instead couples subthreshold membrane voltages:

Experiment 3.3 則是 coupling subthreshold membrane voltages：

§17Parameter counts and claim boundary參數量與 claim boundary

Established已建立

• establishedM6 nominal factorial state substrate: 720 first-spike permutations.M6 nominal factorial state substrate：720 個 first-spike permutations。
• established5-D direct gradient addressing of all 720 targets.5-D Navigator 直接 gradient-address 全部 720 targets。
• established518,400 / 518,400 ordered source→target transitions, without supplied swap paths.518,400 / 518,400 ordered source→target transitions，不需 supplied swap path。
• establishedEach of I, tau, threshold, resistance independently serves as a six-value actuator.I、tau、threshold、resistance 各自都能獨立成為 6-value actuator。
• establishedVoltage-mediated signed coupling can also control the full M6 target set.voltage-mediated signed coupling 也能控制完整 M6 target set。

Not established尚未建立

• openTrain-once reusable addressing function for unseen targets.train once 後對 unseen target 立即 output parameter 的 reusable addressing function。
• openM>6 optimization scaling.M>6 optimization scaling。
• openRobustness under current/parameter/noise/spike-time perturbations.current / parameter / noise / spike-time perturbation 下的 robustness。
• openJoint-parameter identifiability or biological plasticity rule.joint-parameter identifiability 或 biological plasticity rule。
• openA functional task that actually uses all 720 states.真正 functional task 實際利用全部 720 states。

§18Conclusion and references小結與 references

The snnTorch comparison was checked against the current official documentation surfaced as snnTorch 1.0.0. Software APIs can change; the links above are the authority for future revisions.

snnTorch 比較已依目前官方文件（顯示為 snnTorch 1.0.0）核對。software API 之後可能變動，未來更新以官方 links 為準。