321 lines
18 KiB
Markdown
321 lines
18 KiB
Markdown
# The Biological "Cascade of Failure"
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This model now demonstrates **Metabolic Silencing**, which is a highly consistent biological behavior:
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1. High firing rate → **Vesicle Depletion** (Fast).
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2. High firing rate → **ATP Depletion** (Slow).
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3. Low ATP → **Pump Failure** (JPMCA slows down).
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4. Pump Failure → **Residual Calcium** stays high.
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5. Residual Calcium → **CDI stays active** (The VGCCs lock shut).
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6. **Result:** The synapse stops firing to save itself from excitotoxicity.
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**CASCADE 1 — Vesicle Depletion** appears at `Max_RRP`, `Max_RP`, `p_release_base`, `k_rec_fast/slow`, `stochastic_release`, `map_trace_to_speed`, and the recruitment block in Loop 1. The key annotation explains the asymmetry: `p_release_base * Ca_micro` makes each spike draw *more* vesicles as Ca_micro rises early in a burst — a positive feedback that accelerates the collapse before recruitment can respond.
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**CASCADE 2 — ATP Depletion** is anchored at `Glucose_level` (the root input) and at `compute_astrocyte_metabolic_health` in Loop 3, where it explains that `ATP_level` is the bridge variable that carries minute-scale metabolic state into the millisecond Ca²⁺ world.
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**CASCADE 3 — Pump Failure** is annotated at `k_PMCA`, `k_NCX`, `k_SERCA`, `ATP_half`, and `compute_pump_atp_factor`. The NCX comment explicitly notes its role as a floor-not-rescue — it keeps clearing during failure and enables the auto-reset, but cannot prevent accumulation alone.
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**CASCADE 4 — Residual Ca²⁺** appears at `B_total`, `tau_buffer_rebind`, the `capture_fraction` block, and the buffer recharge lines. The buffer saturation note explains the two-phase dynamic: buffer is protective early but becomes invisible once `B_free → 0`.
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**CASCADE 5 — CDI Lock-out** is annotated at `k_CDI_rise`, `Ca_micro_saturation`, `k_CDI_rec`, and both the rise and recovery lines in Loop 1. The recovery comment spells out the self-locking logic explicitly as a chain.
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**CASCADE 6 — Silence** sits at `effective_conductance` with a timing note showing that mGluR fires first, eCB second, and CDI last but irreversibly. The eCB and mGluR annotations in Loop 2 explain their roles as early partial brakes versus the terminal lock.
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## 5. Model Summary Checklist
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- \[x\] **Timing:** Spans 0.1 ms (AP) to 300,000 ms (Metabolism).
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- \[x\] **Conservation:** Vesicles and Neurotransmitters are conserved through the Gln→RP→RRP→Cleft→Astro loop.
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- \[x\] **Stability:** CDI and mGluR/eCB provide three layers of negative feedback to prevent runaway excitation.
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- \[x\] **Resource Constraints:** ATP and Pool guards prevent physically impossible negative values or infinite accumulation.
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---
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This pseudocode serves as a comprehensive blueprint for a biologically consistent tripartite synapse. It is structured into three nested temporal loops to handle the transition from microseconds to minutes.
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---
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### **Variable Reference Table**
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| Variable | Definition | Scale | Role |
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|---------------|------------------------------------|--------|-----------------------------|
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| **Ca_micro** | Free calcium in the active zone | 0.1 ms | Triggers release and CDI |
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| **B_free** | Available buffer sites (Calbindin) | 0.1 ms | Immediate calcium "sponge" |
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| **N_RRP** | Readily Releasable Pool | 1 ms | Immediate vesicle supply |
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| **N_RP** | Reserve Pool | 100 ms | Long-term vesicle warehouse |
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| **Tr_Ca** | Calcium Trace | 10 ms | Memory of recent activity |
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| **CDI_fac** | Inactivation Factor | 50 ms | Internal negative feedback |
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| **mGluR_pre** | Autoreceptor activation | 500 ms | Cleft-sensing inhibition |
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| **ATP_level** | Metabolic energy state | 1 min | Gates calcium clearance |
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### **The Multi-Scale Engine**
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```python
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# --- SIMULATION CONFIGURATION ---
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# dt = 0.1 ms (Fine-grained step)
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# Total_Time = 300,000 ms (5 minutes)
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while t < Total_Time:
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# ============================================================
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# 1. FINE-GRAINED NEURAL LOOP (Every 0.1ms)
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# Handles: Electrical spikes, Buffering, and Influx
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# ============================================================
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# --- ACTION POTENTIAL & WAVEFORM ---
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if is_AP_active(t):
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# Layered Inhibition logic:
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# CDI (Internal), eCB (Retrograde/Post), mGluR (Autoreceptor/Cleft)
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total_inhibition = (1 - CDI_fac) * (1 - eCB_level) * (1 - mGluR_pre * alpha_mGluR)
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# Calculate Influx via VGCC
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# V_pre_pulse(t) accounts for the finite duration of the spike window
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raw_influx = N_VGCC * total_inhibition * V_pre_pulse(t)
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# --- FAST BUFFERING BEHAVIOR ---
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# Immediate capture of influx by buffer proteins (e.g., Calbindin)
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captured = raw_influx * (B_free / B_total)
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B_free = max(0, B_free - captured)
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Ca_bound += captured
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# Resulting free Calcium that actually reaches the sensors
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Ca_micro += (raw_influx - captured)
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# --- STOCHASTIC RELEASE ---
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if N_RRP > 0:
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# Release probability is a function of Ca_micro
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p_release = compute_stochastic_p(Ca_micro, N_RRP)
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if random_uniform(0, 1) < p_release:
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N_RRP -= 1 # Deplete one vesicle
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Glu_cleft += 1 # Release NT into cleft
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CDI_fac += k_CDI_rise # Increment inactivation per release
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# --- CONTINUOUS CALCIUM CLEARANCE ---
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# NCX (Sodium-Calcium Exchanger) - Fast, gradient driven
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# PMCA (Plasma Membrane Ca-ATPase) - Slow, ATP dependent
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atp_efficiency = ATP_level**2 / (ATP_level**2 + 0.3**2)
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cleared = (k_NCX * Ca_micro) + (k_PMCA * Ca_micro * atp_efficiency)
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Ca_micro = max(0.0, Ca_micro - cleared)
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# --- RECOVERY MECHANISMS ---
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# CDI Recovery: Decay of inactivation as Ca_micro falls
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CDI_fac = max(0.0, CDI_fac - (dt / tau_CDI_rec))
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# Buffer Recovery: Re-release of bound ions into microdomain
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re_release = Ca_bound * (dt / tau_buf_release)
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Ca_bound -= re_release
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Ca_micro += re_release
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B_free = B_total - Ca_bound
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# ============================================================
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# 2. MID-GRAINED INTEGRATION (Every 10ms - 100ms)
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# Handles: Recruitment Traces and Autoreceptor Feedback
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# ============================================================
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if t % 10 == 0:
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# TRACE INTEGRATOR: The memory of recent spikes
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Tr_Ca = update_leaky_integrator(Tr_Ca, Ca_micro, tau_trace)
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# RECRUITMENT LOGIC (RP -> RRP)
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# Recruitment speed (k_rec) scales non-linearly with Tr_Ca
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k_rec = compute_k_rec(Tr_Ca)
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# Apply HARD CAPS and GUARDS:
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# 1. Cannot take more than what is in RP
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# 2. Cannot exceed the ceiling of RRP
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refill_qty = k_rec * N_RP * (Max_RRP - N_RRP)
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refill_qty = max(0, min(refill_qty, N_RP))
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N_RRP += refill_qty
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N_RP -= refill_qty
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# AUTORECEPTOR FEEDBACK: Presynapse sensing its own NT
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mGluR_pre += (Glu_cleft / (Glu_cleft + Km) - mGluR_pre) * (10 / tau_mGluR)
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# ============================================================
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# 3. COARSE-GRAINED METABOLIC LOOP (Every 1s - 1min)
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# Handles: Astrocyte support, eCB Brake, and Sustainability
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# ============================================================
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if t % 1000 == 0:
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# ASTROCYTE GLUTAMATE CLEARANCE
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# Astrocytes clean the cleft; NT is recycled into the Glutamine pool
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cleared_glu = Glu_cleft * EAAT_clearance_rate
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Glu_cleft -= cleared_glu
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Gln_pool += cleared_glu
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# RETROGRADE BRAKE (eCB from Postsynapse)
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# Postsynapse synthesizes eCB based on its own V_post activity
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eCB_level = update_retrograde_brake(V_post_history)
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# METABOLIC REPLENISHMENT
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# Astrocyte health determines ATP; Glutamine refills the Reserve Pool
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ATP_level = compute_atp_from_astro_health(Gln_pool, Metabolic_State)
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# Long-term Refill of the Reserve Pool (The Warehouse)
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N_RP = min(N_RP + (Gln_pool * metabolic_shuttle_rate), Max_RP)
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Gln_pool *= 0.9 # Account for metabolic overhead/loss
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t += dt # Increment simulation time
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```
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---
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### 3. Biological Consistency Summary
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1. **Metabolic Coupling:** The `atp_efficiency` variable creates a physical link between the 5-minute astrocyte clock and the 0.1ms calcium clock. If the astrocyte is exhausted, the pumps fail, and the `CDI_fac` locks the synapse into **silence**.
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2. **Double-Lock Inhibition:** You have two distinct brakes. The `mGluR_pre` is "homosynaptic" (local self-check), while the `eCB_level` is "heterosynaptic" (feedback from the receiver). This prevents runaway excitation and excitotoxicity.
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3. **Conservation of Mass:** Neurotransmitters follow a closed loop: `Gln_pool` -> `N_RP` -> `N_RRP` -> `Glu_cleft` -> `Astrocyte` -> `Gln_pool`.
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4. **Buffer Buffer:** The `B_free` mechanism prevents numerical instability during high-frequency bursts by absorbing excess calcium influx instantly.
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---
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The missing presynaptic behaviors are:
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1. **Ca²⁺ clearance** (PMCA, NCX pumps) — without this, Ca²⁺ accumulates indefinitely
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2. **Calcium buffer proteins** (calbindin, calmodulin) — fast buffering before pumps clear
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3. **VGCC inactivation recovery** — CDI is mentioned but the recovery (de-inactivation) is missing
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4. **RRP hard cap** — the refill logic can overshoot without a ceiling
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5. **Reserve Pool (RP) depletion guard** — refill can go negative
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6. **Presynaptic autoreceptors** (mGluR/CB1R feedback closing the loop from NT in cleft → presynaptic suppression) — separate from eCB which comes from postsynapse
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7. **Spike refractory / AP waveform duration** — the effective window for Ca²⁺ entry is finite
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Let me build an annotated code diagram and then write the enhanced model.Here is a full analysis of every missing loop, followed by the updated code for each section.
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---
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## Missing behaviors and the reasoning behind each
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**Why Ca²⁺ clearance is the most critical gap.** `Ca_micro` currently has no exit route — it only grows. Without PMCA pumps, NCX exchangers, and SERCA (ER uptake), every spike leaves residual Ca²⁺ that accumulates across the simulation and eventually locks the synapse in a permanently over-activated (or CDI-locked) state. The clearance mechanisms also run at different speeds: NCX is fast (tens of ms), PMCA is slower but higher-capacity, SERCA is slowest and stores calcium for later use as an internal buffer.
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**Why Ca²⁺ buffer proteins must precede clearance.** Calbindin and calmodulin bind free Ca²⁺ within microseconds and act as a fast, temporary "sponge". They blunt the initial `Ca_micro` peak, protecting against excess CDI. They also slowly release Ca²⁺ back into the cytosol, which feeds the trace integrator more smoothly. Without buffers, the microdomain pulse is unrealistically sharp.
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**Why CDI recovery closes a loop without itself.** The model already writes `CDI_factor` but never resets it. A VGCC that inactivated on spike N stays inactivated on spike N+1. CDI recovery is simply a decay back toward zero, with a time constant of \~100 ms, driven by Ca²⁺ falling (i.e., it depends on clearance — another reason clearance comes first).
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**Why mGluR autoreceptors are needed.** The eCB pathway is a *retrograde* signal synthesized by the *postsynapse*. But the presynapse also has its own direct cleft-sensing system: presynaptic mGluR2/3 receptors bind glutamate in the cleft and suppress VGCC conductance and cAMP. This is a homosynaptic feedback loop that is entirely local to the presynapse and missing from the current model.
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**Why pool guards matter.** The `refill_amount` calculation can produce `N_RP < 0` if `current_recruitment_rate * N_RP > N_RP`. The `N_RRP` overshoot is subtler but also real: if two slow-loop updates stack before the fast loop consumes RRP, you can exceed `Max_RRP`.
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---
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## Updated code, section by section
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### Loop 1A — Ca²⁺ dynamics (replaces the current `Ca_micro +=` block)
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```python
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# --- PRESYNAPTIC Ca2+ DYNAMICS ---
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if V_pre == 1: # AP arrives
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effective_conductance = N_VGCC * (1 - eCB_level) * (1 - CDI_factor)
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raw_influx = compute_flux(effective_conductance, V_pre_voltage)
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# ADDED: Buffer proteins capture a fraction of influx immediately.
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# Buffering capacity (B_free) depletes on capture, recovers slowly.
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# VARIABLE: B_free – free buffer sites (calbindin/calmodulin)
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# TIMING: rebinds saturated buffer in ~200 ms
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captured = raw_influx * (B_free / B_total) # fraction caught
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B_free = max(0, B_free - captured) # buffer saturates
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Ca_micro += (raw_influx - captured) # only free Ca2+ counts
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# --- ADDED: Ca2+ CLEARANCE (runs every ms, not just on spike) ---
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# Three parallel mechanisms, each with its own rate constant:
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# k_PMCA ~0.03 /ms (plasma membrane Ca-ATPase, ATP-dependent)
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# k_NCX ~0.10 /ms (sodium-calcium exchanger, voltage-sensitive, fast)
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# k_SERCA ~0.01 /ms (ER pump, slowest, fills internal Ca2+ store)
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# ADDED: ATP gates pump speed — shared with metabolic loop below
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pump_scale = compute_pump_atp_factor(ATP_level) # 0→1
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cleared_PMCA = k_PMCA * Ca_micro * pump_scale
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cleared_NCX = k_NCX * Ca_micro # NCX is not ATP-dependent
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cleared_SERCA = k_SERCA * Ca_micro * pump_scale
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Ca_micro -= (cleared_PMCA + cleared_NCX + cleared_SERCA)
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Ca_micro = max(0.0, Ca_micro) # hard floor
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# ADDED: SERCA fills an internal ER store (Ca_ER).
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# This store can be released later (e.g. mGluR activation triggers IP3→ER release).
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# For now it is simply a sink; ER-release can be wired later.
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Ca_ER += cleared_SERCA
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# ADDED: Buffer recharge — captured Ca2+ slowly re-releases back to cytosol,
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# and free buffer sites recover as Ca2+ is extruded.
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# TIMING: tau_buffer_rebind ~200 ms
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Ca_micro += Ca_buffer_bound * dt / tau_buffer_rebind
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Ca_buffer_bound *= (1 - dt / tau_buffer_rebind)
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B_free = B_total - Ca_buffer_bound # bookkeeping
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```
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### Loop 1B — CDI recovery (adds a reset that was missing)
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```python
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# --- CDI INACTIVATION + RECOVERY ---
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# EXISTING: CDI_factor rises with Ca_micro on each spike.
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CDI_factor += map_calcium_to_inactivation(Ca_micro)
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# ADDED: CDI_factor decays back to zero as Ca2+ is cleared.
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# VARIABLE: tau_CDI_recovery ~100 ms
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# LOGIC: Recovery tracks Ca_micro level — low Ca2+ → fast de-inactivation.
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CDI_recovery_rate = k_CDI_rec * (1 - Ca_micro / Ca_micro_saturation)
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CDI_factor = max(0.0, CDI_factor - CDI_recovery_rate * dt)
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```
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### Loop 1C — Pool arithmetic with guards
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```python
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# --- RRP RELEASE (with hard cap) ---
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if N_RRP > 0:
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released_NT = stochastic_release(N_RRP, Ca_micro)
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N_RRP = max(0, N_RRP - released_NT)
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add_NT_to_cleft(released_NT)
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# --- RP → RRP RECRUITMENT (with floor and ceiling guards) ---
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current_recruitment_rate = map_trace_to_speed(Tr_Ca)
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refill_amount = current_recruitment_rate * N_RP * (Max_RRP - N_RRP)
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refill_amount = max(0.0, refill_amount) # ADDED: never negative
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refill_amount = min(refill_amount, N_RP) # ADDED: can't take more than RP holds
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N_RRP = min(N_RRP + refill_amount, Max_RRP) # ADDED: hard ceiling
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N_RP = max(0.0, N_RP - refill_amount) # ADDED: hard floor
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```
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### Loop 2 — mGluR autoreceptor (new, 1 s loop)
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```python
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# --- ADDED: HOMOSYNAPTIC AUTORECEPTOR FEEDBACK ---
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# VARIABLE: mGluR_activation – presynaptic mGluR2/3 occupancy (0→1)
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# TIMING: rises in ~500 ms when NT_cleft is high, decays in ~2 s
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# LOGIC: Directly reduces VGCC conductance AND suppresses cAMP
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# (cAMP pathway gates RRP docking speed — can be added later).
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# This loop is distinct from eCB: it is local, homosynaptic, and faster.
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mGluR_activation += (NT_cleft / (NT_cleft + Km_mGluR) - mGluR_activation) * (dt_slow / tau_mGluR)
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# The suppression factor enters the high-freq loop at Line 1A:
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# effective_conductance = N_VGCC * (1 - eCB_level) * (1 - CDI_factor) * (1 - mGluR_activation * alpha_mGluR)
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#
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# alpha_mGluR: max fractional suppression (~0.4 for mGluR2/3 at physiological concentrations)
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```
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### Loop 3 — ATP dependency on pumps (links metabolic health to Ca²⁺ clearance)
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```python
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# --- ADDED: ATP GATES CA2+ PUMP SPEED ---
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# VARIABLE: ATP_level – normalized 0→1 (computed from astrocyte metabolic health)
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# LOGIC: PMCA and SERCA are ATP-dependent.
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# When ATP_level drops, Ca2+ clearance slows → Ca_micro stays elevated →
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# CDI rises → effective VGCC conductance collapses → synapse silences.
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# This is the realistic metabolic-silence cascade.
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def compute_pump_atp_factor(ATP_level):
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# Hill function: half-maximal at ATP_half = 0.3
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return ATP_level**2 / (ATP_level**2 + ATP_half**2)
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# The slowest metabolic loop already writes ATP_level via compute_astrocyte_metabolic_health().
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# No further wiring needed — pump_scale above picks it up automatically.
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```
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---
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## The closed loop, stated plainly
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Every Ca²⁺ that enters now has exactly one exit: PMCA, NCX, or SERCA. Buffers slow the peak. CDI rises with Ca²⁺ and falls as Ca²⁺ falls — it can no longer lock permanently. The cleft-sensing mGluR autoreceptor gives the presynapse its own, faster brake independent of the postsynapse. Pool arithmetic is bounded on both ends. And ATP depletion now cascades naturally: less ATP → slower pumps → higher residual Ca²⁺ → more CDI → fewer effective VGCCs → silence — which is precisely the metabolic-fatigue endpoint the deep loop was trying to express but couldn't reach without the pump link. |