Delayed latents across multiple groups (mDLAG)#

Delayed latents across multiple groups (mDLAG) is a Bayesian dimensionality reduction method for multi-group time series data. It addresses two challenges:

  1. Identifying network-level interactions

  • How many latent dimensions describe the data

  • Which groups each latent dimension involves

  1. Disentangling concurrent signal flow

  • The direction of signal flow across groups

  • How signals evolve over time and across trials (samples)

Note

mDLAG is under active development. The current implementation provides stubs for the planned API.

Model specification#

Observation model. For group \(m\) at time \(t\) on trial \(n\):

\[\mathbf{y}^{(m)}_{n,t} = \mathbf{C}^{(m)} \mathbf{x}^{(m)}_{n,t} + \mathbf{d}^{(m)} + \boldsymbol{\epsilon}^{(m)}\]

where \(\boldsymbol{\epsilon}^{(m)} \sim \mathcal{N}(\mathbf{0}, \text{diag}(\boldsymbol{\phi}^{(m)})^{-1})\).

This is the same linear-Gaussian structure as GFA, but now indexed by time \(t\) and trial (sample) \(n\). The latents \(\mathbf{x}^{(m)}_{n,t}\) are coupled across groups at each time point.

State model. Unlike GFA’s i.i.d. prior, mDLAG places a Gaussian process prior over latents to capture smooth temporal dynamics:

\[\mathbf{x}_{n,j,:} \sim \mathcal{GP}(0, K_j)\]

Each latent \(j\) has:

  • A timescale \(\tau_j\) controlling smoothness

  • Time delays \(D_j^{(m)}\) describing the lead-lag relationship between groups

Time delays and signal flow#

Time delays are the key distinguishing feature of mDLAG. For each latent \(j\), the relative delay between groups indicates signal flow direction:

\[\Delta D_j = D_j^{(m_2)} - D_j^{(m_1)}\]

  • \(\Delta D_j > 0\): group \(m_1\) leads group \(m_2\)

  • \(\Delta D_j < 0\): group \(m_2\) leads group \(m_1\)

  • \(\Delta D_j = 0\): simultaneous activity

Time delays are continuous-valued—not restricted to discrete time bins. By convention, group 1 is the reference (\(D_j^{(1)} = 0\) for all latents).

Automatic relevance determination#

Like GFA, mDLAG uses ARD priors to automatically determine:

  • The total number of latent dimensions

  • Which groups each latent involves

See Group factor analysis (GFA) for details on how ARD enables automatic dimensionality selection.

References#

Gokcen, E., Jasper, A. I., Xu, A., Kohn, A., Machens, C. K. & Yu, B. M. Uncovering motifs of concurrent signaling across multiple neuronal populations. Advances in Neural Information Processing Systems 36, 34711-34722 (2023). Paper link