Fig 1.
Asymmetry and decision transition threshold in the striatum.
(A) Schematic of the striatal circuit (B) Steady state firing rates of the D1 (blue), D2 (red) MSNs and FSI (green) as a function of cortical inputs as estimated from the linearized mean-field dynamics model of the striatum. Here we consider the ‘multiplicative scenario’ for the extra input the the D1 MSNs. We refer to the crossover point (), where the bias of striatal activity (ΔMSN) changes from D1 to D2 MSNs, as the decision transition threshold ≈ 13 Hz here and marked with the dashed line. The grey and black traces show the firing rates of the D1 and D2 MSNs when they received cortical inputs with the same strength, respectively. This shows that extra input to D1 MSNs is necessary to activate the ‘direct’ pathway. Otherwise, D1 MSNs cannot have higher firing rates than D2 MSNs.
Table 1.
Striatal network parameters.
Fig 2.
Balance of D1 and D2 MSNs firing rates as a function of cortical input rates.
(A) ΔMSN as a function of λctx_d1 and λctx_d2 (cf. Eq 1), scenario-I. Diagonal (marked with white line) represents the case that the two MSN populations receive equal cortical drive (λctx_d1 = λctx_d2, X = 0.0). With these inputs, ΔMSN is always negative. This shows the inherent bias towards D2, due to the asymmetrical connectivity. The area below the diagonal represents the regime of higher input drive to D1 λctx_d1 > λctx_d2, X > 0. Off-diagonal bands represent ΔMSN for a constant difference in cortical input rates (ΔCTX) to the two MSN populations. The dashed line marks the desirable regime of operation for the striatum, in which a systematic increase in the cortical input can reverse the sign of ΔMSN. (B) ΔMSN as a function of ΔCTX to the two MSN populations. For a constant value of ΔCTX, ΔMSN changes depending on λCTX (cf. Eq 1). The oval marks the region where an increase in the cortical input rates for a constant ΔCTX changes the sign of ΔMSN from positive to negative, indicating a higher firing rate of the D2 MSNs. (C) Same as in panel A for the scenario-II (cf. Eq 9). In this scenario we considered JC1 > JC2, therefore, the difference in the drive to D1 and D2 MSNs scales with the difference between JC1 and JC2. Since JC1 > JC2, the diagonal itself lies in a desirable regime, where ΔMSN changes from positive to negative for increasing cortical input rate. (D) Same as in panel B for the scenario-II.
Fig 3.
Mechanism of decision transition threshold.
The mechanism of decision transition threshold is explained analytically. (A) ΔMSN plotted for different values of λCTX and λFSI. The dashed line refers to the case of increasing cortical excitation and feedforward FSI inhibition (as also discussed in Fig 2A). However, ΔMSN changes from positive to negative along the rows (keeping λFSI constant and increasing λCTX) and along the columns (keeping λCTX constant and increasing λFSI). (B) Mechanism for scenario I, ‘additive’ input, Refer to Eqs 4–8. ΔMSN = Inpstr + Inpadd plotted for increasing values of λCTX. (C) ΔMSN = λD1 − λD2 as calculated from Eqs 2, 3. (D) ΔMSN plotted for different values of λCTX and λFSI. The dashed line refers to the case of increasing cortical excitation and feedforward FSI inhibition (as also discussed in Fig 2C). However, ΔMSN changes from positive to negative by increasing λFSI for a constant value of λCTX. (E) ΔMSN = D1eff + D2eff plotted for increasing values of λCTX. Mechanism for scenario II, ‘multiplicative’ input. Refer to Eqs 13–14. (F) ΔMSN = λD1 − λD2 as calculated from Eqs 21, 22.
Fig 4.
Decision transition threshold in the striatum.
Mean firing rates of D1 and D2 MSNs populations for different cortical input rates with constant difference between the drives to the two populations (scenario - II cf. Eq 9). The DTT in this case is ≈ 20 Hz. The rasters and PSTH for the spiking activity for D1 and D2 MSNs are shown for two points, before and after crossing the DTT. (A) (top) Rasters for D1 and D2 populations at λCTX = 10.0Hz. (Middle) PSTHs for D1 and D2. (Bottom) Difference between the firing rates of D1 and D2 (i.e ΔMSN). Notice that the firing rate of D1 MSNs is higher than that of D2 MSNs (ΔMSN > 0). At the decision transition threshold, the bias switches from D1 to D2.(B) Same as (A) but at λCTX ≈ 25Hz, where ΔMSN < 0.
Fig 5.
Network activity of D1 and D2 MSNs for selected values of B and W.
In all examples the striatum network was configured according to the scenario-II and D1 MSNs received inputs with higher strength than D2 MSNS. Both MSN populations received cortical input at 7 Hz. (A) Scheme describing the two types of input correlations: W refers to correlations among the pre-synaptic neurons of a single MSN. B′ refers to correlation among the pre-synaptic neurons of two different MSNs.(B)B′ = 0.01, W = 0.01, raster(top) and population activity (bottom). For these inputs the activity of the D1 and D2 MSNS is uncorrelated and, therefore, exerts less effective inhibition on the other population. Because D1 MSNs are configured in the scenario-II, D1 MSNs have a higher average firing rate. (C) B′ = 0.01, W = 0.33. This value falls in the range of W < Wopt. Here D1 operates in the ‘spike-wasting’ regime and, hence, cannot effectively inhibit D2, in spite of a higher drive from cortex. (D) B′ = 0.21, W = 0.33. High values of B′ and W leads to wasted inhibition due to periods of large correlated activity, followed by long periods of quiescence. (E) B′ = 0.21, W = 0.01. For W < Wopt, B increases ΔMSN.
Fig 6.
Effect of B and W input correlations on the balance of D1 and D2 activity.
(A) ΔMSN as a function B and W when the two MSNs population received cortical input with a firing rate of 7 Hz. Here, we considered the scenario-II and set JC1 = 3.6 nS, JC2 = 3.0 nS. At this input rate and with uncorrelated inputs (B = W = 0), D1 MSNS have higher firing rates. The black contour marks the region beyond which ΔMSN is close to zero. All the values are concentrated below the diagonal, because of the constraint: B ≤ W. ΔMSN varies non-monotonically as a function of W for a constant value of B. Dotted arc marks the Wopt for a given firing rate where ΔMSN is maximal. For W ≥ Wopt, increasing B decreases ΔMSN. For values W < Wopt, increasing B increases ΔMSN. (B) same as in (A), for input rate of 23 Hz. Wopt has shifted to 0.3 as compared to 0.2 in panel A. (C) ΔMSN as a function of W and input firing rate for B′ = 0.01. The space spanned by the input firing rates and W can be divided into three distinct regimes. Low W and high firing rates, ΔMSN is negative(blue colours) and the output of the striatum is biased towards D2 MSNs (‘No-Go’ pathway). In the regime where W ≈ Wopt, D1 MSNs have higher firing rate than D2 MSNs and the striatum output is biased towards the ‘Go’ pathway. The third regime spans across very high values of W in which both D1 and D2 MSNs operate in a spike wasting regime and, therefore, ΔMSN is very low. This regime we define as ‘Region of High-Conflict’ (RHC). Because higher firing rates for D1 and D2 MSNs are observed in non-overlapping regions in the space spanned by input correlation and rates, we argue that the striatum may act as a threshold detector and signal the state of cortical inputs by raising the relative activity of D1 or D2 MSNs over the other, respectively.
Fig 7.
Effect of Dopamine and GPe firing rates on the balance of D1 and D2 activity.
(A) Mean firing rates of D1 MSNs and D2 MSNs populations plotted for different levels of dopamine. Darker shades of blue (red) correspond to D1 (D2) MSN activity for higher levels of dopamine. For lower than normal levels of dopamine, the DTT shifts to the left (from ≈ 19 Hz to ≈ 9 Hz). This decreases the regime with a bias towards D1 MSNs. For higher than normal levels of dopamine, the DTT shifts to right (from ≈ 19 to ≈ 27Hz). This, in turns, increases the regime with a bias towards D1 MSNs. (B) Effect of GPe firing rates on the DTT. Solid lines refer to the normal state of the striatum. GPe inhibits the fast spiking interneurons, shifting the DTT to the right (from ≈ 9 to ≈ 14Hz) (dotted lines). Therefore, the regime with a D2 bias (10 < λCTX < 14, D2 solid red line) now has a bias towards D1 MSNs (blue dashed line) (C) In dopamine depleted conditions, inhibition of fast spiking interneurons via GPe is not able to switch the bias from D2 MSNs to D1 MSNs (compare dashed red and dashed blue lines).
Fig 8.
DTT in striatum with symmetrical FSI projections.
(A) For the ‘additive’ input scenario, the striatum exhibits a DTT, in spite of the symmetrical feedforward inhibition from FSIs, refer to Eqs 15–17. A DTT is encountered not only along the diagonal (i.e increasing cortical excitation as well as feedforward inhibition from FSIs-dashed line), but also for clamped value of FSIs with increasing λCTX. (B) ΔMSN = Inpstr + Inpadd plotted for increasing values of λCTX. (C) ΔMSN = λD1 − λD2 as calculated from Eqs 2, 3. (D) In a ‘multiplicative’ input scenario, however, ΔMSN always remains positive for symmetrical FSI projections along the diagonal. A DTT can be imposed by clamping the FSI rates at a constant value, while increasing the cortical excitation. (E) ΔMSN = D1eff + D2eff + commeff plotted for increasing values of λCTX. (F) ΔMSN = λD1 − λD2 as calculated from Eqs 21, 22.
Fig 9.
Modulation of the DTT in the striatum.
(Left) Each contour encloses an area in the parameter space of input correlation (W) and input firing rates, in which the firing rate of D1 MSNs exceeds that of D2 MSNs. The blue contour represents the healthy state of the striatum. Loss of dopamine results in a decrease in the strength of excitatory inputs to the D1 MSNs and, therefore, the blue contour shrank to the red contour indicating that for a large range of firing rates and correlations D2 MSNs have higher firing rate than D1 MSNs. The green contour on the other hand depicts a state in which there is a high level of dopamine in the striatum (e.g. in PD patients on L-Dopa treatment) because excessive dopamine increases the overall excitability of D1 MSNs, thereby expanding the region in which D1 MSNs have a higher firing rate. We refer to the region in which the firing rates of D1 and D2 MSNs are comparable as a region of high-conflict. (Right) The blue contour is the same as in the left panel. The orange contour shows how the increase in shared correlation (B′) could increase the region of high-conflict (RHC).
Table 2.
Parameters for Eq (24).
Table 3.
Parameters for model neurons in network simulations.
Table 4.
Cortical input to striatal neurons.
Table 5.
Inhibitory input to striatal neurons.
Fig 10.
Generation of within and between correlations.
(A) The correlations are generated by copying spikes twice from the mother process (M). Let Rin be the rate of the mother process. The spikes are copied into a set of children processes (C1 ⋯ CN) with a copy probability B′. This controls the shared input to the postsynaptic population and leads to a rate Rin B′. The spikes are copied for a second time with a copy probability W into a pool of input neurons (C11 ⋯ C1N). Each neuron in the input pool spikes with a rate Rin B′W. The correlation within the input pool can be increased by increasing W, whereas the shared input can be increased by increasing the copy probability B′. (B) B ≤ W. The means of the pairwise correlations measured from randomly chosen 2000 pairs of MSN neurons (B) are plotted against the corresponding W values. The different colors represent different input rates in Hz. Notice that most of the points lie below the diagonal.