# Two-State Dynamic Causal Modelling

Two-state DCM is an extension to the standard neuronal model in DCM for fMRI. Whereas the standard neuronal model in DCM represents the activity in each region as a single quantity, two-state DCM has an inhibitory and excitatory population of neurons in each region. This gives an explicit model of intrinsic connectivity within each region, and was adopted to be more plausible and less constrained than the original model. Two-state DCM imposes positivity constraints - all connections between regions are excitatory, which conforms to the organisation of real cortical hierarchies, where long-range connections are glutamatergic. With these richer dynamics, two-state DCM may provide a better fit to fMRI data. Furthermore, the inhibitory and excitatory populations add stability to the model, allowing the priors on the connections to be relaxed, which also may improve the model's ability to explain the data.

## The Two-State model

The model (see figure) involves recurrent connections between the excitatory (E) and inhibitory (I) populations in each region. There is an excitatory connection from E to I, denoted EI, and an inhibitory connection from I to E, denoted IE. There are inhibitory self-connections on E and I, which are referred to as SE and SI respectively. The connections between regions (EE) link the excitatory populations in each. In its current implementation in SPM12, the values assigned to these connections are either estimated when fitting the model to the data (EE and IE), or have fixed values (EI, SI, SE; see table below). The EE extrinsic connections are based on the off-diagonal of the A-matrix (plus modulatory input B if available), whereas the IE self-inhibitory connections take their value from the diagonal of the A-matrix (plus modulatory input B if available).
| Connection | Description | Value |
| IE | intrinsic inhibitory to excitatory | Estimated |
| EE | extrinsic excitatory to excitatory | Estimated |
| EI | intrinsic excitatory to inhibitory | 1 |
| SI | intrinsic self-inhibition (inhibitory) | 1 |
| SE | intrinsic self-inhibition (excitatory) | 0.5 |

## Interpreting the results

To enable certain connections to always have a positive effect and others to always have a negative effect, the parameters of the connections (A-matrix) and modulatory inputs (B) are log scaling parameters that increase or decrease the prior values.
The between-regions EE (excitatory to excitatory) connection strength is computed as follows:
$EE_{ij}=1/8*exp(J_{ij}(t))$
Where $J_{ij}(t)$ is the connection strength (summed across A and B matrices) between a pair of regions i and j at time t:
$J_{ij}(t)=A_{ij}+B_{ij}*u(t)$
The inhibitory self connections IE are transformed in the same way, but they are always negative, to ensure stability:
$IE_{ij}=-1/8*exp(J_{ij}(t))$
The values in the A- and B-matrices therefore scale the prior connection strength, 1/8Hz. A value of 0 in the A-matrix and 0 in the B-matrix for a between-regions connection would equate to a connection strength of 1/8 * exp(0 + 0) = 1/8Hz. Values of 0 for a self-connection would give -1/8 * exp(0+0) = -1/8Hz.
All this means that in order to inspect the results of model estimation, one should first take the exponential of the A- and B-matrices, i.e. exp(DCM.Ep.A) or exp(DCM.Ep.B). (This is done automatically if using the Review tool in the GUI.) The number which results is a scaling factor, which scales the prior. A value of 1 means no effect, values larger than 1 mean a larger amplitude effect than the prior, and values smaller than 1 mean a smaller amplitude effect than the prior. The values in the C-matrix remain in units of Hz.
Here are some more examples of how to interpret the parameters.

### A-matrix between-region connections

A between-region connection exp(DCM.Ep.A(i,j)) larger than 1 would mean the excitatory influence of region j on region i is larger than the prior (1/8Hz). A value smaller than 1 would mean the excitatory influence is smaller than the prior.

### A-matrix self connections

A self-connection exp(DCM.Ep.A(i,i)) larger than 1 would mean stronger (more negative) self-inhibition in region i than the prior (-1/8Hz). A self-connection smaller than 1 would mean weaker (less negative) self-inhibition in region i than the prior.

### Modulatory (task) effects on the self-connections

An estimated parameter exp(DCM.Ep.B(i,i)) larger than 1 would mean an increase in self-inhibition in region i caused by the task. Whereas, a value smaller than 1 on this connection would mean a decrease in self-inhibition caused by the task.

### Modulatory (task) effects on the between-region connections

An estimated parameter exp(DCM.Ep.B(i,j)) larger than 1 on a between-regions connection would mean an increase in the connection strength from region j to region i caused by the task. A value smaller than 1 would mean a decrease in this connection due to the task.

## Difference between the paper and SPM12

The implementation of the model in SPM12, which is described here, has certain differences from its description in the original scientific paper by Marreiros and colleagues. In the paper, all possible intrinsic connections between the excitatory and inhibitory states are modulated and estimated explicitly. Therefore the matrices A and B have size [2xm,2xn] as opposed to [m,n]. This implementation wasn't adopted in SPM, in order to allow a more straightforward model comparison (BMS) with the other single-state and non-linear DCM options. As described above, the software uses a simplified scheme where an estimated parameter larger than 1 on a self-connection on the B-matrix contributes to the IE intrinsic inhibitory to excitatory connection, and therefore an increase in self-inhibition in that region caused by the task.

## References

* 1 2 Marreiros, A.C.; Kiebel, S.J.; Friston, K.J. (2008). "Dynamic causal modelling for fMRI: A two-state model". NeuroImage. 39 (1): 269–278. doi:10.1016/j.neuroimage.2007.08.019. ISSN 1053-8119.