formula for linear mixed effects model using statsmodels.formula.api

Question

I am trying to fit a linear mixed effects model to test whether participants performing a reaction times test become faster as the test progresses. I am using mixedlm() from statsmodels.formula.api. I want to fit a model in which time is the fixed effect (identified by the 'block' variable, which corresponds to the test block presentation order, ranging 0-5). As random effects I want to have random intercepts and random slopes for each participant (identified by the 'test_session_id' variable).'duration' contains the reaction times to each individual stimulus.

Here is my code below.

import statsmodels.formula.api as smf

smf.mixedlm("duration ~ block + (block | test_session_id)", data = sdmt_df_lme, groups=sdmt_df_lme['test_session_id'])

What I don't understand is what should be put into the 'formula' parameter and what should go into the 're_formula' parameter. The documentation is pretty vague, or maybe I don't have a sufficient understanding of the underlying statistics.

Specifically, when I leave 're_formula' blank, I get the following output:

Model:  MixedLM Dependent Variable: duration
No. Observations:   3348    Method: REML
No. Groups: 62  Scale:  0.0603
Min. group size:    54  Log-Likelihood: -192.8144
Max. group size:    54  Converged:  Yes
Mean group size:    54.0        
Coef.   Std.Err.    z   P>|z|   [0.025  0.975]
Intercept   8.159   0.245   33.266  0.000   7.679   8.640
block   -0.016  0.002   -6.609  0.000   -0.021  -0.012
block | test_session_id -0.000  0.000   -1.524  0.128   -0.000  0.000
Group Var   0.061   0.047   

If instead I specify re_formula = '~ block + (block | test_session_id)', I get the following:

/usr/local/lib/python3.10/dist-packages/statsmodels/regression/mixed_linear_model.py:2238: ConvergenceWarning: The MLE may be on the boundary of the parameter space.
  warnings.warn(msg, ConvergenceWarning)
/usr/local/lib/python3.10/dist-packages/statsmodels/regression/mixed_linear_model.py:2262: ConvergenceWarning: The Hessian matrix at the estimated parameter values is not positive definite.
  warnings.warn(msg, ConvergenceWarning)
Model:  MixedLM Dependent Variable: duration
No. Observations:   3348    Method: REML
No. Groups: 62  Scale:  0.0589
Min. group size:    54  Log-Likelihood: -227.0540
Max. group size:    54  Converged:  Yes
Mean group size:    54.0        
Coef.   Std.Err.    z   P>|z|   [0.025  0.975]
Intercept   8.149   0.880   9.265   0.000   6.425   9.873
block   -0.016  0.003   -5.576  0.000   -0.022  -0.011
block | test_session_id -0.000  0.000   -0.403  0.687   -0.000  0.000
Group Var   0.002                   
Group x block Cov   0.000   0.159               
block Var   0.000   0.000               
Group x block | test_session_id Cov -0.000                  
block x block | test_session_id Cov 0.000   0.000               
block | test_session_id Var 0.000   

Any advice is greatly appreciated. Thanks