Cortical neurons in vivo exhibit significant temporal variability, the source of which is crucial for understanding its computational and behavioural implications [1,2]. However, to match experimental observations, single-cell activity must fluctuate over much longer time scales than typical neural and synaptic time scales [3,4]. Balanced networks generally do not produce such slow fluctuations in firing rates, raising questions about their origin. Some theories attribute them to precise synaptic connectivity [5] according to an alternative scenario slow fluctuations in the rates result from non-ergodic network dynamics due to partially-symmetric synaptic connectivity [6,7]. It is unclear whether such ergodicity breaking occurs in networks of spiking neurons [8] and, if so, whether the resulting dynamical regime reproduces the experimentally-observed features of the cortical activity. To address these questions we study the dynamics of sparsely-connected networks of LIF neurons with arbitrary levels of symmetry, q, in the synaptic connectivity. The adjacency matrix is random at $q=0$ and fully symmetric at $q=1$. This results in low, heterogeneous average activity levels and temporally irregular spike trains, resembling cortical activity (Fig. A-B) [9]. To investigate network ergodicity, we estimate single-neuron firing rates over increasing time intervals starting from different initial membrane voltage distributions (Fig. C). In random networks, differences between firing rate estimates from different initial conditions converge to zero over long time windows, indicating ergodicity. For partially-symmetric networks ($q>0$), the onset of the ergodic regime occurs at longer and longer times. For large values of q, D does not decay even for time windows that are 5 order of magnitudes longer than the membrane time constant; the network dynamics is non-ergodic, at least in a weak sense. The phase diagram (Fig.D) reveals a transition from ergodic to non-ergodic dynamics as q increases, especially at higher synaptic strengths (J0). In the non-ergodic regime, network activity is sparse with many nearly silent neurons, a seen experimentally [10]. Our results support the idea that many features of cortical activity can be explained by non-ergodic network dynamics [6]. This regime allows single-neuron activity levels to vary significantly based on microscopic initial conditions, providing a simple explanation for large trial-to-trial fluctuations observed in vivo.