Reinforcement Positional CMR

Reinforcing first-presentation traces at repetition

Reinforcement Positional CMR (ICMR-Reinf) addresses a key question about repetition: when you see an item again, does it strengthen the original memory trace or create an entirely new one?

This model says: both. When an item repeats, it creates a new positional trace and reinforces the first-presentation trace.

The Mechanism

When item \(k\) is presented for the second time (at position \(j\)), having first appeared at position \(i\):

  1. New trace: A new position-\(j\) trace is created (standard positional encoding)
  2. Reinforcement: The position-\(i\) trace is strengthened

The reinforcement step adds associations linking position \(i\) back to the context present at position \(i\) (the original encoding context).

Why Reinforcement?

Standard positional encoding treats each presentation independently. But empirically:

  • People often preferentially recall the first presentation
  • Repetitions boost memory for the original occurrence
  • The “first presentation advantage” persists even with spaced repetitions

Reinforcement captures the idea that seeing something again reminds you of when you first saw it.

Mathematical Specification

At Repetition (item \(k\) at position \(j\), previously at position \(i\))

1. Create new trace (standard): \[\Delta M^{FC}_{jk} = \gamma \mathbf{p}_j \mathbf{c}_j\] \[\Delta M^{CF}_{kj} = \phi_j \mathbf{c}_j \mathbf{p}_j\]

2. Reinforce first presentation:

Retrieve the original encoding context: \[\mathbf{c}^{orig}_i = M^{FC}_{pre} \mathbf{p}_i\]

Add reinforcement associations: \[\Delta M^{FC}_{ik} = \eta_{mfc} \mathbf{p}_i \mathbf{c}^{orig}_i\] \[\Delta M^{CF}_{ki} = \eta_{mcf} \mathbf{c}^{orig}_i \mathbf{p}_i\]

where \(\eta\) is the reinforcement strength.

Parameters

Parameter Symbol Description
first_pres_reinforcement \(\eta\) Shared reinforcement strength for MFC and MCF
mfc_first_pres_reinforcement \(\eta_{mfc}^{+}\) Additional MFC-specific reinforcement
mcf_first_pres_reinforcement \(\eta_{mcf}^{+}\) Additional MCF-specific reinforcement

Total reinforcement: - MFC: \(\eta_{mfc} = \eta + \eta_{mfc}^{+}\) - MCF: \(\eta_{mcf} = \eta + \eta_{mcf}^{+}\)

Predictions

First Presentation Advantage

With reinforcement (\(\eta > 0\)): - First presentations accumulate strength across repetitions - Later presentations contribute their own traces but also boost the first - Recall favors the first presentation’s context

Lag-CRP from First Presentation

When the first presentation is recalled, transitions go to: - Items near position \(i\) (the first presentation) - Not necessarily items near position \(j\) (the repetition)

Usage

Code 
from jaxcmr.models.reinf_positional_cmr import CMR

params = {
    "encoding_drift_rate": 0.5,
    "start_drift_rate": 0.5,
    "recall_drift_rate": 0.5,
    "learning_rate": 0.5,
    "primacy_scale": 2.0,
    "primacy_decay": 0.8,
    "shared_support": 0.05,
    "item_support": 0.25,
    "choice_sensitivity": 0.6,
    "mfc_sensitivity": 3.0,
    "first_pres_reinforcement": 0.5,  # Shared reinforcement
    "mfc_first_pres_reinforcement": 0.0,  # MFC-specific (optional)
    "mcf_first_pres_reinforcement": 0.0,  # MCF-specific (optional)
    "stop_probability_scale": 0.05,
    "stop_probability_growth": 0.2,
    "learn_after_context_update": True,
    "allow_repeated_recalls": False,
}

model = CMR(list_length=16, parameters=params)

Implementation

Key implementation detail:

Code 
def experience_item(self, item_index):
    # Check if item was previously studied
    item_studied = self.studied == item_index + 1
    already_studied = jnp.any(item_studied)

    # Get original encoding context (from pre-experimental MFC)
    associated_context = self.pre_exp_mfc.probe(item_studied)

    # Reinforce first presentation if repetition
    new_mfc = lax.cond(
        already_studied,
        lambda: self.mfc.associate(
            item_studied, associated_context, self.mfc_first_pres_reinforcement
        ),
        lambda: self.mfc,
    )
    new_mcf = lax.cond(
        already_studied,
        lambda: self.mcf.associate(
            associated_context, item_studied, self.mcf_first_pres_reinforcement
        ),
        lambda: self.mcf,
    )

    # Then create new trace at current position...

Comparison: No Reinstatement vs Reinforcement

Model At repetition Benefit of repetition
Standard CMR Blended context Averaged trace
Positional CMR New trace only Multiple traces
No Reinstate CMR New trace, no context blend Distinct traces
Reinforcement CMR New trace + boost first First trace strengthened

Theoretical Connections

This model relates to:

  • Encoding variability theory: Each presentation creates a new trace
  • Remindings hypothesis: Re-seeing something triggers memory of the original
  • Strength-based models: Repetition increases trace strength

The key insight is that reinforcement targets the original context, not the current context—connecting the repetition back to its first occurrence.