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neuron/appunti/2026-03-16-nuovo-modello-tripartita-semplificato.md

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+## The Biological "Cascade of Failure"
+
+This model now demonstrates **Metabolic Silencing**, which is a highly consistent biological behavior:
+
+1. High firing rate → **Vesicle Depletion** (Fast).
+2. High firing rate → **ATP Depletion** (Slow).
+3. Low ATP → **Pump Failure** (JPMCA​ slows down).
+4. Pump Failure → **Residual Calcium** stays high.
+5. Residual Calcium → **CDI stays active** (The VGCCs lock shut).
+6. **Result:** The synapse stops firing to save itself from excitotoxicity.
+
+**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.
+
+**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.
+
+**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.
+
+**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`.
+
+**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.
+
+**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.
+
+## 5. Model Summary Checklist
+
+- \[x\] **Timing:** Spans 0.1 ms (AP) to 300,000 ms (Metabolism).
+- \[x\] **Conservation:** Vesicles and Neurotransmitters are conserved through the Gln→RP→RRP→Cleft→Astro loop.
+- \[x\] **Stability:** CDI and mGluR/eCB provide three layers of negative feedback to prevent runaway excitation.
+- \[x\] **Resource Constraints:** ATP and Pool guards prevent physically impossible negative values or infinite accumulation.
+
+---
+
+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.
+
+---
+
+### **Variable Reference Table**
+
+| Variable      | Definition                         | Scale  | Role                        |
+|---------------|------------------------------------|--------|-----------------------------|
+| **Ca_micro**  | Free calcium in the active zone    | 0.1 ms | Triggers release and CDI    |
+| **B_free**    | Available buffer sites (Calbindin) | 0.1 ms | Immediate calcium "sponge"  |
+| **N_RRP**     | Readily Releasable Pool            | 1 ms   | Immediate vesicle supply    |
+| **N_RP**      | Reserve Pool                       | 100 ms | Long-term vesicle warehouse |
+| **Tr_Ca**     | Calcium Trace                      | 10 ms  | Memory of recent activity   |
+| **CDI_fac**   | Inactivation Factor                | 50 ms  | Internal negative feedback  |
+| **mGluR_pre** | Autoreceptor activation            | 500 ms | Cleft-sensing inhibition    |
+| **ATP_level** | Metabolic energy state             | 1 min  | Gates calcium clearance     |
+
+### **The Multi-Scale Engine**
+
+```python
+# --- SIMULATION CONFIGURATION ---
+# dt = 0.1 ms (Fine-grained step)
+# Total_Time = 300,000 ms (5 minutes)
+
+while t < Total_Time:
+
+    # ============================================================
+    # 1. FINE-GRAINED NEURAL LOOP (Every 0.1ms)
+    # Handles: Electrical spikes, Buffering, and Influx
+    # ============================================================
+    
+    # --- ACTION POTENTIAL & WAVEFORM ---
+    if is_AP_active(t):
+        # Layered Inhibition logic:
+        # CDI (Internal), eCB (Retrograde/Post), mGluR (Autoreceptor/Cleft)
+        total_inhibition = (1 - CDI_fac) * (1 - eCB_level) * (1 - mGluR_pre * alpha_mGluR)
+        
+        # Calculate Influx via VGCC
+        # V_pre_pulse(t) accounts for the finite duration of the spike window
+        raw_influx = N_VGCC * total_inhibition * V_pre_pulse(t)
+        
+        # --- FAST BUFFERING BEHAVIOR ---
+        # Immediate capture of influx by buffer proteins (e.g., Calbindin)
+        captured = raw_influx * (B_free / B_total)
+        B_free = max(0, B_free - captured)
+        Ca_bound += captured
+        
+        # Resulting free Calcium that actually reaches the sensors
+        Ca_micro += (raw_influx - captured)
+        
+        # --- STOCHASTIC RELEASE ---
+        if N_RRP > 0:
+            # Release probability is a function of Ca_micro
+            p_release = compute_stochastic_p(Ca_micro, N_RRP)
+            if random_uniform(0, 1) < p_release:
+                N_RRP -= 1             # Deplete one vesicle
+                Glu_cleft += 1         # Release NT into cleft
+                CDI_fac += k_CDI_rise  # Increment inactivation per release
+
+    # --- CONTINUOUS CALCIUM CLEARANCE ---
+    # NCX (Sodium-Calcium Exchanger) - Fast, gradient driven
+    # PMCA (Plasma Membrane Ca-ATPase) - Slow, ATP dependent
+    atp_efficiency = ATP_level**2 / (ATP_level**2 + 0.3**2)
+    cleared = (k_NCX * Ca_micro) + (k_PMCA * Ca_micro * atp_efficiency)
+    Ca_micro = max(0.0, Ca_micro - cleared)
+    
+    # --- RECOVERY MECHANISMS ---
+    # CDI Recovery: Decay of inactivation as Ca_micro falls
+    CDI_fac = max(0.0, CDI_fac - (dt / tau_CDI_rec))
+    
+    # Buffer Recovery: Re-release of bound ions into microdomain
+    re_release = Ca_bound * (dt / tau_buf_release)
+    Ca_bound -= re_release
+    Ca_micro += re_release
+    B_free = B_total - Ca_bound
+
+    # ============================================================
+    # 2. MID-GRAINED INTEGRATION (Every 10ms - 100ms)
+    # Handles: Recruitment Traces and Autoreceptor Feedback
+    # ============================================================
+    if t % 10 == 0:
+        # TRACE INTEGRATOR: The memory of recent spikes
+        Tr_Ca = update_leaky_integrator(Tr_Ca, Ca_micro, tau_trace)
+        
+        # RECRUITMENT LOGIC (RP -> RRP)
+        # Recruitment speed (k_rec) scales non-linearly with Tr_Ca
+        k_rec = compute_k_rec(Tr_Ca) 
+        
+        # Apply HARD CAPS and GUARDS:
+        # 1. Cannot take more than what is in RP
+        # 2. Cannot exceed the ceiling of RRP
+        refill_qty = k_rec * N_RP * (Max_RRP - N_RRP)
+        refill_qty = max(0, min(refill_qty, N_RP))
+        
+        N_RRP += refill_qty
+        N_RP -= refill_qty
+
+        # AUTORECEPTOR FEEDBACK: Presynapse sensing its own NT
+        mGluR_pre += (Glu_cleft / (Glu_cleft + Km) - mGluR_pre) * (10 / tau_mGluR)
+
+    # ============================================================
+    # 3. COARSE-GRAINED METABOLIC LOOP (Every 1s - 1min)
+    # Handles: Astrocyte support, eCB Brake, and Sustainability
+    # ============================================================
+    if t % 1000 == 0:
+        # ASTROCYTE GLUTAMATE CLEARANCE
+        # Astrocytes clean the cleft; NT is recycled into the Glutamine pool
+        cleared_glu = Glu_cleft * EAAT_clearance_rate
+        Glu_cleft -= cleared_glu
+        Gln_pool += cleared_glu
+        
+        # RETROGRADE BRAKE (eCB from Postsynapse)
+        # Postsynapse synthesizes eCB based on its own V_post activity
+        eCB_level = update_retrograde_brake(V_post_history)
+        
+        # METABOLIC REPLENISHMENT
+        # Astrocyte health determines ATP; Glutamine refills the Reserve Pool
+        ATP_level = compute_atp_from_astro_health(Gln_pool, Metabolic_State)
+        
+        # Long-term Refill of the Reserve Pool (The Warehouse)
+        N_RP = min(N_RP + (Gln_pool * metabolic_shuttle_rate), Max_RP)
+        Gln_pool *= 0.9 # Account for metabolic overhead/loss
+
+    t += dt # Increment simulation time
+```
+
+---
+
+### 3. Biological Consistency Summary
+
+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**.
+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.
+3. **Conservation of Mass:** Neurotransmitters follow a closed loop: `Gln_pool` -> `N_RP` -> `N_RRP` -> `Glu_cleft` -> `Astrocyte` -> `Gln_pool`.
+4. **Buffer Buffer:** The `B_free` mechanism prevents numerical instability during high-frequency bursts by absorbing excess calcium influx instantly.
+
+---
+
+The missing presynaptic behaviors are:
+
+1. **Ca²⁺ clearance** (PMCA, NCX pumps) — without this, Ca²⁺ accumulates indefinitely
+2. **Calcium buffer proteins** (calbindin, calmodulin) — fast buffering before pumps clear
+3. **VGCC inactivation recovery** — CDI is mentioned but the recovery (de-inactivation) is missing
+4. **RRP hard cap** — the refill logic can overshoot without a ceiling
+5. **Reserve Pool (RP) depletion guard** — refill can go negative
+6. **Presynaptic autoreceptors** (mGluR/CB1R feedback closing the loop from NT in cleft → presynaptic suppression) — separate from eCB which comes from postsynapse
+7. **Spike refractory / AP waveform duration** — the effective window for Ca²⁺ entry is finite
+
+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.
+
+![image.png](.attachments.1175009/image.png)
+
+---
+
+## Missing behaviors and the reasoning behind each
+
+**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.
+
+**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.
+
+**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).
+
+**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.
+
+**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`.
+
+---
+
+## Updated code, section by section
+
+### Loop 1A — Ca²⁺ dynamics (replaces the current `Ca_micro +=` block)
+
+```python
+# --- PRESYNAPTIC Ca2+ DYNAMICS ---
+
+if V_pre == 1:  # AP arrives
+    effective_conductance = N_VGCC * (1 - eCB_level) * (1 - CDI_factor)
+    raw_influx = compute_flux(effective_conductance, V_pre_voltage)
+    
+    # ADDED: Buffer proteins capture a fraction of influx immediately.
+    # Buffering capacity (B_free) depletes on capture, recovers slowly.
+    # VARIABLE: B_free  – free buffer sites (calbindin/calmodulin)
+    # TIMING: rebinds saturated buffer in ~200 ms
+    captured = raw_influx * (B_free / B_total)   # fraction caught
+    B_free   = max(0, B_free - captured)          # buffer saturates
+    Ca_micro += (raw_influx - captured)            # only free Ca2+ counts
+
+# --- ADDED: Ca2+ CLEARANCE (runs every ms, not just on spike) ---
+# Three parallel mechanisms, each with its own rate constant:
+# k_PMCA  ~0.03 /ms  (plasma membrane Ca-ATPase, ATP-dependent)
+# k_NCX   ~0.10 /ms  (sodium-calcium exchanger, voltage-sensitive, fast)
+# k_SERCA ~0.01 /ms  (ER pump, slowest, fills internal Ca2+ store)
+
+# ADDED: ATP gates pump speed — shared with metabolic loop below
+pump_scale = compute_pump_atp_factor(ATP_level)   # 0→1
+
+cleared_PMCA  = k_PMCA  * Ca_micro * pump_scale
+cleared_NCX   = k_NCX   * Ca_micro               # NCX is not ATP-dependent
+cleared_SERCA = k_SERCA * Ca_micro * pump_scale
+
+Ca_micro -= (cleared_PMCA + cleared_NCX + cleared_SERCA)
+Ca_micro  = max(0.0, Ca_micro)                    # hard floor
+
+# ADDED: SERCA fills an internal ER store (Ca_ER).
+# This store can be released later (e.g. mGluR activation triggers IP3→ER release).
+# For now it is simply a sink; ER-release can be wired later.
+Ca_ER += cleared_SERCA
+
+# ADDED: Buffer recharge — captured Ca2+ slowly re-releases back to cytosol,
+# and free buffer sites recover as Ca2+ is extruded.
+# TIMING: tau_buffer_rebind ~200 ms
+Ca_micro += Ca_buffer_bound * dt / tau_buffer_rebind
+Ca_buffer_bound *= (1 - dt / tau_buffer_rebind)
+B_free = B_total - Ca_buffer_bound                # bookkeeping
+```
+
+### Loop 1B — CDI recovery (adds a reset that was missing)
+
+```python
+# --- CDI INACTIVATION + RECOVERY ---
+# EXISTING: CDI_factor rises with Ca_micro on each spike.
+CDI_factor += map_calcium_to_inactivation(Ca_micro)
+
+# ADDED: CDI_factor decays back to zero as Ca2+ is cleared.
+# VARIABLE: tau_CDI_recovery ~100 ms
+# LOGIC: Recovery tracks Ca_micro level — low Ca2+ → fast de-inactivation.
+CDI_recovery_rate = k_CDI_rec * (1 - Ca_micro / Ca_micro_saturation)
+CDI_factor = max(0.0, CDI_factor - CDI_recovery_rate * dt)
+```
+
+### Loop 1C — Pool arithmetic with guards
+
+```python
+# --- RRP RELEASE (with hard cap) ---
+if N_RRP > 0:
+    released_NT = stochastic_release(N_RRP, Ca_micro)
+    N_RRP = max(0, N_RRP - released_NT)
+    add_NT_to_cleft(released_NT)
+
+# --- RP → RRP RECRUITMENT (with floor and ceiling guards) ---
+current_recruitment_rate = map_trace_to_speed(Tr_Ca)
+refill_amount = current_recruitment_rate * N_RP * (Max_RRP - N_RRP)
+refill_amount = max(0.0, refill_amount)           # ADDED: never negative
+refill_amount = min(refill_amount, N_RP)          # ADDED: can't take more than RP holds
+
+N_RRP = min(N_RRP + refill_amount, Max_RRP)       # ADDED: hard ceiling
+N_RP  = max(0.0, N_RP  - refill_amount)           # ADDED: hard floor
+```
+
+### Loop 2 — mGluR autoreceptor (new, 1 s loop)
+
+```python
+# --- ADDED: HOMOSYNAPTIC AUTORECEPTOR FEEDBACK ---
+# VARIABLE: mGluR_activation  – presynaptic mGluR2/3 occupancy (0→1)
+# TIMING: rises in ~500 ms when NT_cleft is high, decays in ~2 s
+# LOGIC: Directly reduces VGCC conductance AND suppresses cAMP
+#        (cAMP pathway gates RRP docking speed — can be added later).
+# This loop is distinct from eCB: it is local, homosynaptic, and faster.
+
+mGluR_activation += (NT_cleft / (NT_cleft + Km_mGluR) - mGluR_activation) * (dt_slow / tau_mGluR)
+
+# The suppression factor enters the high-freq loop at Line 1A:
+# effective_conductance = N_VGCC * (1 - eCB_level) * (1 - CDI_factor) * (1 - mGluR_activation * alpha_mGluR)
+#
+# alpha_mGluR: max fractional suppression (~0.4 for mGluR2/3 at physiological concentrations)
+```
+
+### Loop 3 — ATP dependency on pumps (links metabolic health to Ca²⁺ clearance)
+
+```python
+# --- ADDED: ATP GATES CA2+ PUMP SPEED ---
+# VARIABLE: ATP_level  – normalized 0→1 (computed from astrocyte metabolic health)
+# LOGIC: PMCA and SERCA are ATP-dependent.
+#        When ATP_level drops, Ca2+ clearance slows → Ca_micro stays elevated →
+#        CDI rises → effective VGCC conductance collapses → synapse silences.
+#        This is the realistic metabolic-silence cascade.
+
+def compute_pump_atp_factor(ATP_level):
+    # Hill function: half-maximal at ATP_half = 0.3
+    return ATP_level**2 / (ATP_level**2 + ATP_half**2)
+
+# The slowest metabolic loop already writes ATP_level via compute_astrocyte_metabolic_health().
+# No further wiring needed — pump_scale above picks it up automatically.
+```
+
+---
+
+## The closed loop, stated plainly
+
+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.