Top-down Control of Inhibition Reshapes Neural Dynamics Giving Rise to a Diversity of Computations

Growing evidence shows that top-down projections from excitatory neurons in higher brain areas selectively synapse onto local inhibitory interneurons in sensory systems. While this connectivity is conserved across sensory modalities, the role of this feedback in shaping the dynamics of local circuits, and the resultant computational benefits it provides remains poorly understood. Using rate models of neuronal firing in a network consisting of excitatory, inhibitory and top-down populations, we found that changes in the weight of feedback to inhibitory neurons generated diverse network dynamics and complex transitions between these dynamics. Additionally, modulation of the weight of top-down feedback supported a number of computations, including both pattern separation and oscillatory synchrony. A bifurcation analysis of the network identified a new mechanism by which gamma oscillations could be generated in a model of neural circuits, which we termed Top-down control of Inhibitory Neuron Gamma (TING). We identified the unique roles that top-down feedback of inhibition plays in shaping network dynamics and computation, and the ways in which these dynamics can be deployed to process sensory inputs. Significance Statement The functional role of feedback projections, connecting excitatory neurons in higher brain areas to inhibitory neurons in primary sensory regions, remains a fundamental open question in neuroscience. Growing evidence suggests that this architecture is recapitulated across a diverse array of sensory systems, ranging from vision to olfaction. Using a rate model of top-down feedback onto inhibition, we found that changes in the weight of feedback support both pattern separation and oscillatory synchrony, including a mechanism by which top-down inputs could entrain gamma oscillations within local networks. These dual functions were accomplished via a codimension-2 bifurcation in the dynamical system. Our results highlight a key role for this top-down feedback, gating inhibition to facilitate often diametrically different local computations.


Introduction
However, for the purpose of pattern separation, the two metrics did not give 148 qualitatively different results when assessing the distances due to changes the 149 feedback weight 46 (Fig.S4, Supplementary materials). Additionally, we found 150 that the non-monotonic dependence of distance in both L and N on the feedback 151 weight ( 46 WXY ) persisted, and that an optimal value for any given pair of stimuli 152 could be found (Fig.S5, Supplementary materials).

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Bifurcation analysis 155 We denoted the dynamical system of Eqn.
(1) as a parameterized form 156 ̇= ( , ) All the other parameters were fixed as constants determined based on previous 164 experimental work (Whittington, Traub et al. 2000). 165 As the system Eqn.  If none of the eigenvalues of ( d , d ) lied on the imaginary axis (i.e., the (where all three populations were silent). For small stimuli, the trajectory made an 234 excursion before spiraling into an equilibrium indicated by the solid dot (Fig.1D, 235 orange). Similarly, when the stimulus was large, the trajectory again settled into 236 an equilibrium, but one that was translated within the phase space to the top-right 237 corner (Fig.1D, black). Finally, medium stimuli, the time-varying oscillation of the 238 firing rates manifested as a periodic orbit (or a limit cycle) in the 3D phase space 239 (Fig.1D, brown). In this space, we defined the steady-state dynamics as the -limit 240 set of the system. with the steady-state dynamics transitioning to a periodic orbit (a limit cycle).

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Changing the weight of top-down projections onto the local inhibitory population (I) 265 in a single stimulus produced the same diversity of firing rate dynamics that 266 occurred from changes in the stimulus. Furthermore, for a given top-down input 267 weight ( 46 ), the effects on the network dynamics for a particular stimulus was 268 unique to that stimulus ( Fig.2B  To test this hypothesis, we presented our network with a pair of stimuli, 298 denoted by . and 4 (corresponding to stimuli arranged along a one-dimensional 299 axis) and studied how top-down control of inhibition altered the firing rate 300 representations of the two stimuli (Fig.3A). For a set of stimuli " , = 1,2, the 301 steady-state representation of the network activity was the ω-limit set Ω " , = 1,2 302 for the associated stimulus with the distance between the two stimuli . and 4 Δ 303 in the stimulus space, and the resultant distance in the firing rate phase space 304 between the two -limit sets (Ω . and Ω 4 ) as a metric (see Methods  for one stimulus while that balance was preserved for the second stimulus. optimal for both synchrony between the oscillations generated by stimulus . and 384 representations of stimulus . and 6 . A 3D scatter plot of . − 4 − 6 ( Fig.4D), 385 showed all examples corresponding to two pairs of stimuli: ( . , 4 ) whose 386 oscillatory responses to stimuli were synchronous for a given 46 and two stimuli 387 ( . , 6 ) whose responses were also maximally separable at the same 46 . The 388 values of top-down input 46 for each point that fulfilled these diametrically distinct 389 functions were coded by color and size (Fig.4E). Importantly, we found that the

Stimulus of intermediate strength evoked oscillations of firing rates 25
Our network model generated diverse dynamics in response to different stimuli . As the 26 input increased from low to high values, the steady states of the dynamics transitioned 27 from equilibria to limit cycles and then back to equilibria again (Fig. 1). We plotted firing 28 rates of the top-down population (T) after transient dynamics as a function of input 29 magnitude (Fig S1). An intermediate interval existed where # occupied a continuum of 30 firing rates for one stimulus, indicating the emergence of oscillations. However, when the 31 stimulus strength was beyond that interval, either too low or too high, the firing rate # 32 achieved a constant value for the stimulus. corresponding to the top-down projection to inhibitory neurons modulated the oscillations 37 over a range of frequencies as well as amplitudes (Fig. 2). The control of both the 38 frequency and amplitude via changes in )# occurred across an array of weights ( #+ ) 39 associated the feedforward drive from the local excitatory population (E) to the feedback 40 population (Fig.S2). Interestingly, the range of )# within which neural oscillations 41 emerged became narrower with increasing #+ , suggesting that the feedforward drive 42 established the dynamic range within which changes in feedback ( )# ) adjusted both the 43 amplitude and frequency of the oscillations that could be generated by the network.
to the oscillation of firing rates in the temporal space. Since the time constants in our 50 model had units of millisecond, the frequency of oscillations was defined as 1000⁄ , with 51 T denoting the period of limit cycles, which was defined as follows: if 0 ( ), = 1,2,3 denote 52 the firing rate of neural population in the model, then a limit cycle satisfies the periodicity: 53 0 ( ) = 0 ( + ), = 1,2,3 54 for some T>0 and all ∈ ℝ. The minimal T for which the above equality holds is the period 55 of the limit cycle. 56 57

Definition of Euclidean distance and spectrum distance 58
Here we defined two types of distances, the Euclidean distance > and the spectrum 59 representations reflected larger distances, that would improve the discriminability of saddle point and the nullplane (Fig.S7B) corresponding to the second equation of