Guiding Synchrony through Random Networks

Sparse random networks contain structures that can be considered as diluted feed-forward networks. Modeling of cortical circuits has shown that feed-forward structures, if strongly pronounced compared to the embedding random network, enable reliable signal transmission by propagating localized (subnetwork) synchrony. This assumed prominence, however, is not experimentally observed in local cortical circuits. Here, we show that nonlinear dendritic interactions, as discovered in recent single-neuron experiments, naturally enable guided synchrony propagation already in random recurrent neural networks that exhibit mildly enhanced, biologically plausible substructures.

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Memory, thought, and language: It has been hypothesized that the essence of these functions of our brain is the coordinated propagation and transformation of synchronous activity of nerve cells. How can synchronous neuronal activity propagate in cortical networks where the constituent neurons in their functionally base states send short electrical pulses (spikes) in a seemingly random fashion? An established hypothesis is that there exist “synfire chains” in our cortical networks, structures that guide directional propagation of neuronal activity by feeding the synchronous spiking of a group of neurons to a subsequent group of neurons that fire synchronously following the input—so the process continues. Previous modeling studies had to assume very prominent structures—in terms of how strongly and how often neurons in these structures couple to each other—to obtain robust propagation. Such prominent synfire chains, however, have not been found experimentally. This theoretical work suggests that a nonadditivity in the neuronal coupling may render prominent structural synfire chains unnecessary for guided propagation of synchronous neuronal activity.

Each neuron receives inputs from many others and fires in response to the sum of the received inputs. The traditional theoretical description of neuronal coupling assumes that the summation is linear, i.e., a simple addition of all the received inputs. Recently, however, single-neuron experiments have revealed that neurons are capable of fast, nonlinear summation of synchronously received inputs. We have incorporated this nonlinear mechanism into both a minimal and a biologically more detailed model of cortical networks. By investigating the models both analytically and numerically, we have found that the nonadditive enhancement in neuronal coupling reduces the need for dense or strong structural, anatomy-based coupling, and ultimately leads to propagation of synchrony guided by weakly feed-forward structures that occur naturally in cortical networks.

We believe that the incorporation of mechanisms of nonadditivity represents a new direction for exploring guided synchrony. The theoretical tools we have developed in this work should also apply to settings that involve non-neuronal networks, e.g., networks of interacting flashing fireflies.

Figure 1

Example dynamics of a conductance-based, leaky integrate-and-fire neuron with dendritic spike generation. (The neuron is initially at resting membrane potential $V^{\mathrm{rest}}=-65\mathrm{mV}$, there are no external inputs, and $I^0=0$.) Panel (a) shows the pEPSP after a stimulation versus the expected pEPSP, i.e., the pEPSP for a neuron without dendritic spike generation. For inputs corresponding to a pEPSP larger than about $3.8\mathrm{mV}$, a dendritic spike is generated which leads to a higher depolarization than expected from additive integration. Panel (b) shows the time course of the membrane potential of a neuron with (green) and without (black) nonlinear dendritic interaction in response to different excitatory inputs sequences. (The red horizontal line indicates the somatic spike threshold). Panel (c) shows the input sequences (black lines, strength: $g^{\mathrm{ex}}=2.3\mathrm{nS}$; close-ups given in insets) and the sum $g_{l,\Delta t}(t)$ of excitatory inputs received within the dendritic integration window [$t-\Delta t$, $t$] (gray lines); cf. Eq. (9). At the first spike arrival around $t=1\mathrm{ms}$, three inputs are received within $\Delta t$ such that $g_{l,\Delta t}(t)$ reaches 6.9 nS. The sum is smaller than the dendritic threshold $g^{\Theta }=8.65\mathrm{nS}$ [red horizontal line in (c)], so no dendritic spike is generated and there is no difference between the membrane potential for a neuron with and without a mechanism for dendritic spike generation. Around $t=50\mathrm{ms}$, four spikes arrive within $\Delta t$, $g_{l,\Delta t}(t)$ exceeds the dendritic threshold, and a dendritic spike is generated. Around $t=100\mathrm{ms}$, four spikes arrive at the neuron, but the temporal difference between the last and the first spike is slightly larger than $\Delta t$. Consequently, $g_{l,\Delta t}(t)$ does not exceed the dendritic threshold and no dendritic spike is initiated.

Figure 2

Emergence of propagation of synchrony. (a) Analytically derived iterated maps approximating the time evolution of the synchronous pulse [solid line, cf. Eq. (16)] and transition probability obtained from network simulations (color code). (b) The basin of attraction of the stable fixed point $G_2$ is illustrated. Initial pulses within the range $(G_1,\omega ]$ propagate with an average pulse size around $G_2$. (c) Iterated maps for FFNs with linear (solid lines) and nonlinear dendritic interactions (dashed lines). Nonlinear interactions reduce the connectivity required for propagation and allow for smaller fractions of active neurons.

Figure 3

Critical connectivity in isolated FFNs. (a),(b) Network simulations (symbols) agree well with analytical predictions (solid lines) (30, 45). The critical connectivity decays with layer size and coupling strength. (c) The reduction factor $c=p_{\mathrm{L}}^{*}/p_{\mathrm{NL}}^{*}>1$ shows that nonlinear dendritic interaction compensates for reduced connectivity. In both scenarios, with linear and nonlinear coupling, we find that $p^{*}\propto {\omega }^{-1}$, such that the reduction factor is independent of the layer size. In networks with linear couplings, the critical connectivity is $p^{*}\propto {ϵ}^{-1}$, whereas, in networks with nonlinear coupling, the dependence on ${ϵ}^{-1}$ is nonlinear. Therefore, the reduction factor increases with decreasing coupling strength. Dashed horizontal lines indicate jumps in the reduction factor at which the number of inputs needed for dendritic spike generation changes.

Figure 4

Nonlinear dendritic interactions enable guided propagation of synchrony in random networks ($N=10000$, $p=0.03$, $\omega =30$). (a),(b)  Detection probability of induced synchronous activity in (a) conventional networks and (b) in networks with nonlinear dendrites. Insets of (a),(b) and Figs. (c),(d)_Typical examples of network activity for different synaptic-enhancement factors. Red and blue indicate spikes of neurons within the FFN; green indicates spikes of neurons randomly sampled from the remaining neurons.